Why is $lim_{xto -infty}sqrt{x^2+5x+3}+x = lim_{xto -infty}(-x)+x=lim_{xto -infty}0 = 0$ not correct?
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$lim_{xto -infty}sqrt{x^2+5x+3}+x = lim_{xto -infty}(-x)+x=lim_{xto -infty}0 = 0$
Apparently, the 2nd step is illegal here. Probably because for $x=-infty$ I'd get $(+infty-infty)$ which is not possible. I see why this wouldn't be possible, I'm not sure if it really is the cause which makes that equation illegal though.
But now, apparently, I could do this:
$lim_{xto -infty}sqrt{x^2+5x+3}+x=lim_{xto -infty}frac{[sqrt{x^2+5x+3}+x][sqrt{x^2+5x+3}-x]}{sqrt{x^2+5x+3}-x}=lim_{xto -infty}frac{x^2+5x+3-x^2}{sqrt{x^2+5x+3}-x}=lim_{xto -infty}frac{5x+3}{sqrt{x^2+5x+3}-x}=lim_{xto -infty}frac{5+3/x}{sqrt{1+5/x+3/x^2}-1}=-5/2$
which gives me the correct result.
But in the 3th step I used $x^2-x^2=0$, how is that legal?
Also, in the 2nd step I implicitly used:
$-xsqrt{x^2+5x+3}+xsqrt{x^2+5x+3}=0$
Which also seems to be fine, but why?
calculus limits
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up vote
7
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$lim_{xto -infty}sqrt{x^2+5x+3}+x = lim_{xto -infty}(-x)+x=lim_{xto -infty}0 = 0$
Apparently, the 2nd step is illegal here. Probably because for $x=-infty$ I'd get $(+infty-infty)$ which is not possible. I see why this wouldn't be possible, I'm not sure if it really is the cause which makes that equation illegal though.
But now, apparently, I could do this:
$lim_{xto -infty}sqrt{x^2+5x+3}+x=lim_{xto -infty}frac{[sqrt{x^2+5x+3}+x][sqrt{x^2+5x+3}-x]}{sqrt{x^2+5x+3}-x}=lim_{xto -infty}frac{x^2+5x+3-x^2}{sqrt{x^2+5x+3}-x}=lim_{xto -infty}frac{5x+3}{sqrt{x^2+5x+3}-x}=lim_{xto -infty}frac{5+3/x}{sqrt{1+5/x+3/x^2}-1}=-5/2$
which gives me the correct result.
But in the 3th step I used $x^2-x^2=0$, how is that legal?
Also, in the 2nd step I implicitly used:
$-xsqrt{x^2+5x+3}+xsqrt{x^2+5x+3}=0$
Which also seems to be fine, but why?
calculus limits
What's the issue with $x^2-x^2=0$? It's a perfectly legal statement. One shouldn't doubt oneself too much. On the other hand the issue with your first attempt is that you can't replace $sqrt{x^2+5x+3}$ with $-x$ as they are not equal. One can only replace one expression with another expression when both are equal.
– Paramanand Singh
Dec 4 at 3:57
Technically, your first two inequalities are not wrong because all terms equal 0.
– Dirk
Dec 4 at 5:55
How did you even make that huge leap of faith to go from $sqrt{x^2+5x+3}$ to $-x$? Even if you made $sqrt{x^2+5x+3}$ become $x$, although wrong, this would still be understandable, but where did the $-$ pop up from?
– YiFan
Dec 5 at 2:55
@YiFan: you should note that $x$ is negative and square root is always non-negative so $-x$ is a reasonable choice.
– Paramanand Singh
Dec 5 at 3:34
add a comment |
up vote
7
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up vote
7
down vote
favorite
$lim_{xto -infty}sqrt{x^2+5x+3}+x = lim_{xto -infty}(-x)+x=lim_{xto -infty}0 = 0$
Apparently, the 2nd step is illegal here. Probably because for $x=-infty$ I'd get $(+infty-infty)$ which is not possible. I see why this wouldn't be possible, I'm not sure if it really is the cause which makes that equation illegal though.
But now, apparently, I could do this:
$lim_{xto -infty}sqrt{x^2+5x+3}+x=lim_{xto -infty}frac{[sqrt{x^2+5x+3}+x][sqrt{x^2+5x+3}-x]}{sqrt{x^2+5x+3}-x}=lim_{xto -infty}frac{x^2+5x+3-x^2}{sqrt{x^2+5x+3}-x}=lim_{xto -infty}frac{5x+3}{sqrt{x^2+5x+3}-x}=lim_{xto -infty}frac{5+3/x}{sqrt{1+5/x+3/x^2}-1}=-5/2$
which gives me the correct result.
But in the 3th step I used $x^2-x^2=0$, how is that legal?
Also, in the 2nd step I implicitly used:
$-xsqrt{x^2+5x+3}+xsqrt{x^2+5x+3}=0$
Which also seems to be fine, but why?
calculus limits
$lim_{xto -infty}sqrt{x^2+5x+3}+x = lim_{xto -infty}(-x)+x=lim_{xto -infty}0 = 0$
Apparently, the 2nd step is illegal here. Probably because for $x=-infty$ I'd get $(+infty-infty)$ which is not possible. I see why this wouldn't be possible, I'm not sure if it really is the cause which makes that equation illegal though.
But now, apparently, I could do this:
$lim_{xto -infty}sqrt{x^2+5x+3}+x=lim_{xto -infty}frac{[sqrt{x^2+5x+3}+x][sqrt{x^2+5x+3}-x]}{sqrt{x^2+5x+3}-x}=lim_{xto -infty}frac{x^2+5x+3-x^2}{sqrt{x^2+5x+3}-x}=lim_{xto -infty}frac{5x+3}{sqrt{x^2+5x+3}-x}=lim_{xto -infty}frac{5+3/x}{sqrt{1+5/x+3/x^2}-1}=-5/2$
which gives me the correct result.
But in the 3th step I used $x^2-x^2=0$, how is that legal?
Also, in the 2nd step I implicitly used:
$-xsqrt{x^2+5x+3}+xsqrt{x^2+5x+3}=0$
Which also seems to be fine, but why?
calculus limits
calculus limits
edited Dec 4 at 8:17
Jeel Shah
5,238105398
5,238105398
asked Dec 3 at 14:28
xotix
36829
36829
What's the issue with $x^2-x^2=0$? It's a perfectly legal statement. One shouldn't doubt oneself too much. On the other hand the issue with your first attempt is that you can't replace $sqrt{x^2+5x+3}$ with $-x$ as they are not equal. One can only replace one expression with another expression when both are equal.
– Paramanand Singh
Dec 4 at 3:57
Technically, your first two inequalities are not wrong because all terms equal 0.
– Dirk
Dec 4 at 5:55
How did you even make that huge leap of faith to go from $sqrt{x^2+5x+3}$ to $-x$? Even if you made $sqrt{x^2+5x+3}$ become $x$, although wrong, this would still be understandable, but where did the $-$ pop up from?
– YiFan
Dec 5 at 2:55
@YiFan: you should note that $x$ is negative and square root is always non-negative so $-x$ is a reasonable choice.
– Paramanand Singh
Dec 5 at 3:34
add a comment |
What's the issue with $x^2-x^2=0$? It's a perfectly legal statement. One shouldn't doubt oneself too much. On the other hand the issue with your first attempt is that you can't replace $sqrt{x^2+5x+3}$ with $-x$ as they are not equal. One can only replace one expression with another expression when both are equal.
– Paramanand Singh
Dec 4 at 3:57
Technically, your first two inequalities are not wrong because all terms equal 0.
– Dirk
Dec 4 at 5:55
How did you even make that huge leap of faith to go from $sqrt{x^2+5x+3}$ to $-x$? Even if you made $sqrt{x^2+5x+3}$ become $x$, although wrong, this would still be understandable, but where did the $-$ pop up from?
– YiFan
Dec 5 at 2:55
@YiFan: you should note that $x$ is negative and square root is always non-negative so $-x$ is a reasonable choice.
– Paramanand Singh
Dec 5 at 3:34
What's the issue with $x^2-x^2=0$? It's a perfectly legal statement. One shouldn't doubt oneself too much. On the other hand the issue with your first attempt is that you can't replace $sqrt{x^2+5x+3}$ with $-x$ as they are not equal. One can only replace one expression with another expression when both are equal.
– Paramanand Singh
Dec 4 at 3:57
What's the issue with $x^2-x^2=0$? It's a perfectly legal statement. One shouldn't doubt oneself too much. On the other hand the issue with your first attempt is that you can't replace $sqrt{x^2+5x+3}$ with $-x$ as they are not equal. One can only replace one expression with another expression when both are equal.
– Paramanand Singh
Dec 4 at 3:57
Technically, your first two inequalities are not wrong because all terms equal 0.
– Dirk
Dec 4 at 5:55
Technically, your first two inequalities are not wrong because all terms equal 0.
– Dirk
Dec 4 at 5:55
How did you even make that huge leap of faith to go from $sqrt{x^2+5x+3}$ to $-x$? Even if you made $sqrt{x^2+5x+3}$ become $x$, although wrong, this would still be understandable, but where did the $-$ pop up from?
– YiFan
Dec 5 at 2:55
How did you even make that huge leap of faith to go from $sqrt{x^2+5x+3}$ to $-x$? Even if you made $sqrt{x^2+5x+3}$ become $x$, although wrong, this would still be understandable, but where did the $-$ pop up from?
– YiFan
Dec 5 at 2:55
@YiFan: you should note that $x$ is negative and square root is always non-negative so $-x$ is a reasonable choice.
– Paramanand Singh
Dec 5 at 3:34
@YiFan: you should note that $x$ is negative and square root is always non-negative so $-x$ is a reasonable choice.
– Paramanand Singh
Dec 5 at 3:34
add a comment |
7 Answers
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To avoid confusion with sign, in these cases I suggest to take $y=-xto infty$ then
$$lim_{xto -infty}sqrt{x^2+5x+3}+x =lim_{yto infty}sqrt{y^2-5y+3}-y $$
from here we can clearly see that the expression is an indeterminate form $infty-infty$ then we can proceed by
$$sqrt{y^2-5y+3}-y=left(sqrt{y^2-5y+3}-yright)cdot frac{sqrt{y^2-5y+3}+y}{sqrt{y^2-5y+3}+y}$$
Note that your first cancellation is not correct because you have considered $sqrt{x^2+5x+3}=-x$ which is not true whereas in the second case we have cancelled out $x^2-x^2$ which is a correct step.
2
I mean that's a nice hint but I can't see how that answers y question. There isn't much difference between $lim_{xto -infty} x-x$ vs. $lim_{yto infty}-y+y$
– xotix
Dec 3 at 14:35
@xotix I've added a specific observation about your wrong step.
– gimusi
Dec 3 at 14:43
add a comment |
up vote
10
down vote
$lim_{xto -infty}sqrt{x^2+5x+3}+x = lim_{xto -infty}(-x)+x$
Apparently, the 2nd step is illegal here.
In this step, you dropped $5x+3$ to go from $sqrt{x^2}$ to $-x$, but what makes you think you can just leave out the terms $5x+3$...? Surely you agree that $x^2+5x+3 ne x^2$.
But in the 3th step I used $x^2-x^2=0$, how is that legal?
How could that not be legal? The difference of two equal real numbers is $0$, of course!
The same goes for:
Also, in the 2nd step I implicitly used:
$-xsqrt{x^2+5x+3}+xsqrt{x^2+5x+3}=0$
Addition after comment: the idea of dominating terms (in a polynomial) is fine, but you cannot choose to apply the limit "locally" and leave the other terms unchanged.
To clarify, this is fine:
$$lim_{x to -infty}sqrt{x^2+5x+3}=lim_{x to -infty}sqrt{x^2}=+infty$$
but that's not what we have here; we have:
$$lim_{x to -infty}left(sqrt{x^2+5x+3}color{blue}{+x}right)$$
and we cannot take the limit of the terms separately
$$lim_{x to -infty}left(sqrt{x^2+5x+3}color{blue}{+x}right)color{red}{ne}lim_{x to -infty}sqrt{x^2+5x+3}+lim_{x to -infty} x$$
because both limits need to exist in order for this step to be valid.
The idea behind $lim_{xto -infty}sqrt{x^2+5x+3}=lim_{xto -infty}sqrt{x^2}=lim_{xto -infty} |x| = lim_{xto -infty} -x$ was, that $x^2$ dominantes $5x+3$. I also thought if that argument is wrong here, if $lim_{x to -infty} x-x=0$ is legal and everything of the same "type"; then I'm okay again. Then I have to think about why the domination argument doesn't work as I thought it does here.
– xotix
Dec 3 at 14:39
@xotix I added a part; the problem lies in taking the limit(s) separately/"locally".
– StackTD
Dec 3 at 14:45
add a comment |
up vote
4
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The first step you took simply wasn’t correct.
$$lim_{x to -infty}sqrt{x^2+5x+3}+x color{red}{neq lim_{x to infty}-x+x}$$
You shouldn’t split a limit if the individual limits don’t exist or are undefined. Going for $infty-infty$ is certainly incorrect as it’s an indeterminate form, so you should always avoid it.
As for your question, you’re cancelling out $x^2$ and $-x^2$. That’s simple algebra and has nothing to do with whether $x$ is approaching an infinite value or not. It always applies for all additive inverses.
add a comment |
up vote
2
down vote
You may set $x = -frac{1}{t}$ and consider the limit for $stackrel{trightarrow 0+}{longrightarrow}$:
$$begin{eqnarray*} sqrt{x^2+5x+3}+x
& stackrel{x = -frac{1}{t}}{=} & frac{sqrt{1-5t +3t^2} - 1}{t} \
& stackrel{trightarrow 0+}{longrightarrow} & f'(0) = -frac{5}{2}mbox{ for } f(t) = sqrt{1-5t +3t^2}
end{eqnarray*}$$
add a comment |
up vote
2
down vote
The first step is illegal, not the second. This is because $sqrt{x^2 + 5x + 3} notequiv -x$.
Also addressing your comment:
The idea behind $lim_{x to -infty} sqrt{x^2 + 5x + 3} = dots = lim_{x to -infty} −x$ was, that $x^2$ dominantes $5x+3$. I also thought if that argument is wrong here, if $lim_{x to -infty} x − x = 0$ is legal and everything of the same "type"; then I'm okay again. Then I have to think about why the domination argument doesn't work as I thought it does here.
You are right in thinking that $x^2$ dominates $5x+3$, so that $color{blue}{sqrt{x^2 + 5x + 3}} sim color{red}{-x}$ for large negative $x$. Indeed there is no problem when computing their ratio:
$$lim_{x to -infty} frac{color{blue}{sqrt{x^2 + 5x + 3}}}{color{red}{-x}} = 1$$
However, because you are computing the difference between $color{blue}{sqrt{x^2 + 5x + 3}}$ and $color{red}{-x}$, such an approximation is not fine enough. You need more terms of the series expansion:
$$
begin{align*}
color{blue}{sqrt{x^2 + 5x + 3}}
&= -x sqrt{1 + frac{5}{x} + frac{3}{x^2}} \
&sim -x left(1 + frac{1}{2} cdot frac{5}{x} + Oleft(frac{1}{x^2}right) right) \
&= color{red}{-x} - frac{5}{2} + Oleft(frac{1}{x}right)
end{align*}
$$
Then may you conclude that the required limit is $-5/2$.
By analogy, an approximation such as $color{blue}{x + 1} sim color{red}{x}$ is perfectly fine for computing the ratio limit $lim_{x to -infty} (color{blue}{x + 1}) / (color{red}{x})$. But if you are computing the difference limit, then you may not discard the $1$ in $color{blue}{x + 1}$.
add a comment |
up vote
1
down vote
Possibly the biggest problem in your first paragraph is that $sqrt{text{whatever}}$ is non-negative when we are working over the reals. Can you exhibit any choice of $x$ in the real numbers giving a negative value of $sqrt{x^2 + 5 x + 3}$?
add a comment |
up vote
0
down vote
I will flesh out much of what would be considered a series of valid steps you may take in order to get something similar to the first step performed in the initial limit calculation. I will note that the second step that was taken is valid because we perform all of the algebra in the limit expression first, then we take a limit of what results.
The following line of reasoning shows that whenever you have $sqrt{x^2+beta x+alpha}$ as a term (added or substracted, not multiplied or divided) in a limit, it may be replaced freely by $|x+frac{beta}{2}|$ inside the limit (of course slight adjustments must be made for it to be shown in general).
Note that for every $b>frac{5}{2}$,
$$lim_{xto-infty}sqrt{x^2+2bx+b^2}+x lelim_{xto-infty}sqrt{x^2+5x+3}+x.$$
To see why this is the case, here is a series of inequalities. First pick a $b>frac{5}{2}$. Then pick $x$ so that $xlefrac{3-b^2}{2b-5}$.
Then
$$(2b-5)xle 3-b^2,$$ $$2bx+b^2le 5x+3,$$
$$x^2+2bx+b^2le x^2+5x+3,$$
$$sqrt{x^2+2bx+b^2}lesqrt{x^2+5x+3},$$
$$sqrt{x^2+2bx+b^2}+xlesqrt{x^2+5x+3}+x,$$
$$lim_{xto-infty}sqrt{x^2+2bx+b^2}+x lelim_{xto-infty}sqrt{x^2+5x+3}+x.$$
The intuition for this comes largely from working backward. So thus the claim is justified.
So now we can say that
$$lim_{bto frac{5}{2}^-}lim_{xto-infty}sqrt{x^2+2bx+b^2}+x lelim_{xto-infty}sqrt{x^2+5x+3}+x,$$
$$lim_{bto frac{5}{2}^-}|x+b|+x=lim_{bto frac{5}{2}^-}-x-b+x=lim_{bto frac{5}{2}^-}-b=-frac{5}{2}lelim_{xto-infty}sqrt{x^2+5x+3}+x.$$
Now since $3<frac{25}{4}$, we have that.
$$lim_{xto-infty}sqrt{x^2+5x+3}+xle lim_{xto-infty}sqrt{x^2+5x+frac{25}{4}}+x=lim_{xto-infty}|x+frac{5}{2}|+x=lim_{xto-infty}-x-frac{5}{2}+x=lim_{xto-infty}-frac{5}{2}=-frac{5}{2}.$$
So $$lim_{xto-infty}sqrt{x^2+5x+3}+x=-frac{5}{2}.$$
add a comment |
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7 Answers
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7 Answers
7
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1
down vote
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To avoid confusion with sign, in these cases I suggest to take $y=-xto infty$ then
$$lim_{xto -infty}sqrt{x^2+5x+3}+x =lim_{yto infty}sqrt{y^2-5y+3}-y $$
from here we can clearly see that the expression is an indeterminate form $infty-infty$ then we can proceed by
$$sqrt{y^2-5y+3}-y=left(sqrt{y^2-5y+3}-yright)cdot frac{sqrt{y^2-5y+3}+y}{sqrt{y^2-5y+3}+y}$$
Note that your first cancellation is not correct because you have considered $sqrt{x^2+5x+3}=-x$ which is not true whereas in the second case we have cancelled out $x^2-x^2$ which is a correct step.
2
I mean that's a nice hint but I can't see how that answers y question. There isn't much difference between $lim_{xto -infty} x-x$ vs. $lim_{yto infty}-y+y$
– xotix
Dec 3 at 14:35
@xotix I've added a specific observation about your wrong step.
– gimusi
Dec 3 at 14:43
add a comment |
up vote
1
down vote
accepted
To avoid confusion with sign, in these cases I suggest to take $y=-xto infty$ then
$$lim_{xto -infty}sqrt{x^2+5x+3}+x =lim_{yto infty}sqrt{y^2-5y+3}-y $$
from here we can clearly see that the expression is an indeterminate form $infty-infty$ then we can proceed by
$$sqrt{y^2-5y+3}-y=left(sqrt{y^2-5y+3}-yright)cdot frac{sqrt{y^2-5y+3}+y}{sqrt{y^2-5y+3}+y}$$
Note that your first cancellation is not correct because you have considered $sqrt{x^2+5x+3}=-x$ which is not true whereas in the second case we have cancelled out $x^2-x^2$ which is a correct step.
2
I mean that's a nice hint but I can't see how that answers y question. There isn't much difference between $lim_{xto -infty} x-x$ vs. $lim_{yto infty}-y+y$
– xotix
Dec 3 at 14:35
@xotix I've added a specific observation about your wrong step.
– gimusi
Dec 3 at 14:43
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
To avoid confusion with sign, in these cases I suggest to take $y=-xto infty$ then
$$lim_{xto -infty}sqrt{x^2+5x+3}+x =lim_{yto infty}sqrt{y^2-5y+3}-y $$
from here we can clearly see that the expression is an indeterminate form $infty-infty$ then we can proceed by
$$sqrt{y^2-5y+3}-y=left(sqrt{y^2-5y+3}-yright)cdot frac{sqrt{y^2-5y+3}+y}{sqrt{y^2-5y+3}+y}$$
Note that your first cancellation is not correct because you have considered $sqrt{x^2+5x+3}=-x$ which is not true whereas in the second case we have cancelled out $x^2-x^2$ which is a correct step.
To avoid confusion with sign, in these cases I suggest to take $y=-xto infty$ then
$$lim_{xto -infty}sqrt{x^2+5x+3}+x =lim_{yto infty}sqrt{y^2-5y+3}-y $$
from here we can clearly see that the expression is an indeterminate form $infty-infty$ then we can proceed by
$$sqrt{y^2-5y+3}-y=left(sqrt{y^2-5y+3}-yright)cdot frac{sqrt{y^2-5y+3}+y}{sqrt{y^2-5y+3}+y}$$
Note that your first cancellation is not correct because you have considered $sqrt{x^2+5x+3}=-x$ which is not true whereas in the second case we have cancelled out $x^2-x^2$ which is a correct step.
edited Dec 3 at 14:42
answered Dec 3 at 14:31
gimusi
91.7k74495
91.7k74495
2
I mean that's a nice hint but I can't see how that answers y question. There isn't much difference between $lim_{xto -infty} x-x$ vs. $lim_{yto infty}-y+y$
– xotix
Dec 3 at 14:35
@xotix I've added a specific observation about your wrong step.
– gimusi
Dec 3 at 14:43
add a comment |
2
I mean that's a nice hint but I can't see how that answers y question. There isn't much difference between $lim_{xto -infty} x-x$ vs. $lim_{yto infty}-y+y$
– xotix
Dec 3 at 14:35
@xotix I've added a specific observation about your wrong step.
– gimusi
Dec 3 at 14:43
2
2
I mean that's a nice hint but I can't see how that answers y question. There isn't much difference between $lim_{xto -infty} x-x$ vs. $lim_{yto infty}-y+y$
– xotix
Dec 3 at 14:35
I mean that's a nice hint but I can't see how that answers y question. There isn't much difference between $lim_{xto -infty} x-x$ vs. $lim_{yto infty}-y+y$
– xotix
Dec 3 at 14:35
@xotix I've added a specific observation about your wrong step.
– gimusi
Dec 3 at 14:43
@xotix I've added a specific observation about your wrong step.
– gimusi
Dec 3 at 14:43
add a comment |
up vote
10
down vote
$lim_{xto -infty}sqrt{x^2+5x+3}+x = lim_{xto -infty}(-x)+x$
Apparently, the 2nd step is illegal here.
In this step, you dropped $5x+3$ to go from $sqrt{x^2}$ to $-x$, but what makes you think you can just leave out the terms $5x+3$...? Surely you agree that $x^2+5x+3 ne x^2$.
But in the 3th step I used $x^2-x^2=0$, how is that legal?
How could that not be legal? The difference of two equal real numbers is $0$, of course!
The same goes for:
Also, in the 2nd step I implicitly used:
$-xsqrt{x^2+5x+3}+xsqrt{x^2+5x+3}=0$
Addition after comment: the idea of dominating terms (in a polynomial) is fine, but you cannot choose to apply the limit "locally" and leave the other terms unchanged.
To clarify, this is fine:
$$lim_{x to -infty}sqrt{x^2+5x+3}=lim_{x to -infty}sqrt{x^2}=+infty$$
but that's not what we have here; we have:
$$lim_{x to -infty}left(sqrt{x^2+5x+3}color{blue}{+x}right)$$
and we cannot take the limit of the terms separately
$$lim_{x to -infty}left(sqrt{x^2+5x+3}color{blue}{+x}right)color{red}{ne}lim_{x to -infty}sqrt{x^2+5x+3}+lim_{x to -infty} x$$
because both limits need to exist in order for this step to be valid.
The idea behind $lim_{xto -infty}sqrt{x^2+5x+3}=lim_{xto -infty}sqrt{x^2}=lim_{xto -infty} |x| = lim_{xto -infty} -x$ was, that $x^2$ dominantes $5x+3$. I also thought if that argument is wrong here, if $lim_{x to -infty} x-x=0$ is legal and everything of the same "type"; then I'm okay again. Then I have to think about why the domination argument doesn't work as I thought it does here.
– xotix
Dec 3 at 14:39
@xotix I added a part; the problem lies in taking the limit(s) separately/"locally".
– StackTD
Dec 3 at 14:45
add a comment |
up vote
10
down vote
$lim_{xto -infty}sqrt{x^2+5x+3}+x = lim_{xto -infty}(-x)+x$
Apparently, the 2nd step is illegal here.
In this step, you dropped $5x+3$ to go from $sqrt{x^2}$ to $-x$, but what makes you think you can just leave out the terms $5x+3$...? Surely you agree that $x^2+5x+3 ne x^2$.
But in the 3th step I used $x^2-x^2=0$, how is that legal?
How could that not be legal? The difference of two equal real numbers is $0$, of course!
The same goes for:
Also, in the 2nd step I implicitly used:
$-xsqrt{x^2+5x+3}+xsqrt{x^2+5x+3}=0$
Addition after comment: the idea of dominating terms (in a polynomial) is fine, but you cannot choose to apply the limit "locally" and leave the other terms unchanged.
To clarify, this is fine:
$$lim_{x to -infty}sqrt{x^2+5x+3}=lim_{x to -infty}sqrt{x^2}=+infty$$
but that's not what we have here; we have:
$$lim_{x to -infty}left(sqrt{x^2+5x+3}color{blue}{+x}right)$$
and we cannot take the limit of the terms separately
$$lim_{x to -infty}left(sqrt{x^2+5x+3}color{blue}{+x}right)color{red}{ne}lim_{x to -infty}sqrt{x^2+5x+3}+lim_{x to -infty} x$$
because both limits need to exist in order for this step to be valid.
The idea behind $lim_{xto -infty}sqrt{x^2+5x+3}=lim_{xto -infty}sqrt{x^2}=lim_{xto -infty} |x| = lim_{xto -infty} -x$ was, that $x^2$ dominantes $5x+3$. I also thought if that argument is wrong here, if $lim_{x to -infty} x-x=0$ is legal and everything of the same "type"; then I'm okay again. Then I have to think about why the domination argument doesn't work as I thought it does here.
– xotix
Dec 3 at 14:39
@xotix I added a part; the problem lies in taking the limit(s) separately/"locally".
– StackTD
Dec 3 at 14:45
add a comment |
up vote
10
down vote
up vote
10
down vote
$lim_{xto -infty}sqrt{x^2+5x+3}+x = lim_{xto -infty}(-x)+x$
Apparently, the 2nd step is illegal here.
In this step, you dropped $5x+3$ to go from $sqrt{x^2}$ to $-x$, but what makes you think you can just leave out the terms $5x+3$...? Surely you agree that $x^2+5x+3 ne x^2$.
But in the 3th step I used $x^2-x^2=0$, how is that legal?
How could that not be legal? The difference of two equal real numbers is $0$, of course!
The same goes for:
Also, in the 2nd step I implicitly used:
$-xsqrt{x^2+5x+3}+xsqrt{x^2+5x+3}=0$
Addition after comment: the idea of dominating terms (in a polynomial) is fine, but you cannot choose to apply the limit "locally" and leave the other terms unchanged.
To clarify, this is fine:
$$lim_{x to -infty}sqrt{x^2+5x+3}=lim_{x to -infty}sqrt{x^2}=+infty$$
but that's not what we have here; we have:
$$lim_{x to -infty}left(sqrt{x^2+5x+3}color{blue}{+x}right)$$
and we cannot take the limit of the terms separately
$$lim_{x to -infty}left(sqrt{x^2+5x+3}color{blue}{+x}right)color{red}{ne}lim_{x to -infty}sqrt{x^2+5x+3}+lim_{x to -infty} x$$
because both limits need to exist in order for this step to be valid.
$lim_{xto -infty}sqrt{x^2+5x+3}+x = lim_{xto -infty}(-x)+x$
Apparently, the 2nd step is illegal here.
In this step, you dropped $5x+3$ to go from $sqrt{x^2}$ to $-x$, but what makes you think you can just leave out the terms $5x+3$...? Surely you agree that $x^2+5x+3 ne x^2$.
But in the 3th step I used $x^2-x^2=0$, how is that legal?
How could that not be legal? The difference of two equal real numbers is $0$, of course!
The same goes for:
Also, in the 2nd step I implicitly used:
$-xsqrt{x^2+5x+3}+xsqrt{x^2+5x+3}=0$
Addition after comment: the idea of dominating terms (in a polynomial) is fine, but you cannot choose to apply the limit "locally" and leave the other terms unchanged.
To clarify, this is fine:
$$lim_{x to -infty}sqrt{x^2+5x+3}=lim_{x to -infty}sqrt{x^2}=+infty$$
but that's not what we have here; we have:
$$lim_{x to -infty}left(sqrt{x^2+5x+3}color{blue}{+x}right)$$
and we cannot take the limit of the terms separately
$$lim_{x to -infty}left(sqrt{x^2+5x+3}color{blue}{+x}right)color{red}{ne}lim_{x to -infty}sqrt{x^2+5x+3}+lim_{x to -infty} x$$
because both limits need to exist in order for this step to be valid.
edited Dec 4 at 8:56
answered Dec 3 at 14:34
StackTD
21.8k1946
21.8k1946
The idea behind $lim_{xto -infty}sqrt{x^2+5x+3}=lim_{xto -infty}sqrt{x^2}=lim_{xto -infty} |x| = lim_{xto -infty} -x$ was, that $x^2$ dominantes $5x+3$. I also thought if that argument is wrong here, if $lim_{x to -infty} x-x=0$ is legal and everything of the same "type"; then I'm okay again. Then I have to think about why the domination argument doesn't work as I thought it does here.
– xotix
Dec 3 at 14:39
@xotix I added a part; the problem lies in taking the limit(s) separately/"locally".
– StackTD
Dec 3 at 14:45
add a comment |
The idea behind $lim_{xto -infty}sqrt{x^2+5x+3}=lim_{xto -infty}sqrt{x^2}=lim_{xto -infty} |x| = lim_{xto -infty} -x$ was, that $x^2$ dominantes $5x+3$. I also thought if that argument is wrong here, if $lim_{x to -infty} x-x=0$ is legal and everything of the same "type"; then I'm okay again. Then I have to think about why the domination argument doesn't work as I thought it does here.
– xotix
Dec 3 at 14:39
@xotix I added a part; the problem lies in taking the limit(s) separately/"locally".
– StackTD
Dec 3 at 14:45
The idea behind $lim_{xto -infty}sqrt{x^2+5x+3}=lim_{xto -infty}sqrt{x^2}=lim_{xto -infty} |x| = lim_{xto -infty} -x$ was, that $x^2$ dominantes $5x+3$. I also thought if that argument is wrong here, if $lim_{x to -infty} x-x=0$ is legal and everything of the same "type"; then I'm okay again. Then I have to think about why the domination argument doesn't work as I thought it does here.
– xotix
Dec 3 at 14:39
The idea behind $lim_{xto -infty}sqrt{x^2+5x+3}=lim_{xto -infty}sqrt{x^2}=lim_{xto -infty} |x| = lim_{xto -infty} -x$ was, that $x^2$ dominantes $5x+3$. I also thought if that argument is wrong here, if $lim_{x to -infty} x-x=0$ is legal and everything of the same "type"; then I'm okay again. Then I have to think about why the domination argument doesn't work as I thought it does here.
– xotix
Dec 3 at 14:39
@xotix I added a part; the problem lies in taking the limit(s) separately/"locally".
– StackTD
Dec 3 at 14:45
@xotix I added a part; the problem lies in taking the limit(s) separately/"locally".
– StackTD
Dec 3 at 14:45
add a comment |
up vote
4
down vote
The first step you took simply wasn’t correct.
$$lim_{x to -infty}sqrt{x^2+5x+3}+x color{red}{neq lim_{x to infty}-x+x}$$
You shouldn’t split a limit if the individual limits don’t exist or are undefined. Going for $infty-infty$ is certainly incorrect as it’s an indeterminate form, so you should always avoid it.
As for your question, you’re cancelling out $x^2$ and $-x^2$. That’s simple algebra and has nothing to do with whether $x$ is approaching an infinite value or not. It always applies for all additive inverses.
add a comment |
up vote
4
down vote
The first step you took simply wasn’t correct.
$$lim_{x to -infty}sqrt{x^2+5x+3}+x color{red}{neq lim_{x to infty}-x+x}$$
You shouldn’t split a limit if the individual limits don’t exist or are undefined. Going for $infty-infty$ is certainly incorrect as it’s an indeterminate form, so you should always avoid it.
As for your question, you’re cancelling out $x^2$ and $-x^2$. That’s simple algebra and has nothing to do with whether $x$ is approaching an infinite value or not. It always applies for all additive inverses.
add a comment |
up vote
4
down vote
up vote
4
down vote
The first step you took simply wasn’t correct.
$$lim_{x to -infty}sqrt{x^2+5x+3}+x color{red}{neq lim_{x to infty}-x+x}$$
You shouldn’t split a limit if the individual limits don’t exist or are undefined. Going for $infty-infty$ is certainly incorrect as it’s an indeterminate form, so you should always avoid it.
As for your question, you’re cancelling out $x^2$ and $-x^2$. That’s simple algebra and has nothing to do with whether $x$ is approaching an infinite value or not. It always applies for all additive inverses.
The first step you took simply wasn’t correct.
$$lim_{x to -infty}sqrt{x^2+5x+3}+x color{red}{neq lim_{x to infty}-x+x}$$
You shouldn’t split a limit if the individual limits don’t exist or are undefined. Going for $infty-infty$ is certainly incorrect as it’s an indeterminate form, so you should always avoid it.
As for your question, you’re cancelling out $x^2$ and $-x^2$. That’s simple algebra and has nothing to do with whether $x$ is approaching an infinite value or not. It always applies for all additive inverses.
edited Dec 3 at 14:52
answered Dec 3 at 14:43
KM101
3,609417
3,609417
add a comment |
add a comment |
up vote
2
down vote
You may set $x = -frac{1}{t}$ and consider the limit for $stackrel{trightarrow 0+}{longrightarrow}$:
$$begin{eqnarray*} sqrt{x^2+5x+3}+x
& stackrel{x = -frac{1}{t}}{=} & frac{sqrt{1-5t +3t^2} - 1}{t} \
& stackrel{trightarrow 0+}{longrightarrow} & f'(0) = -frac{5}{2}mbox{ for } f(t) = sqrt{1-5t +3t^2}
end{eqnarray*}$$
add a comment |
up vote
2
down vote
You may set $x = -frac{1}{t}$ and consider the limit for $stackrel{trightarrow 0+}{longrightarrow}$:
$$begin{eqnarray*} sqrt{x^2+5x+3}+x
& stackrel{x = -frac{1}{t}}{=} & frac{sqrt{1-5t +3t^2} - 1}{t} \
& stackrel{trightarrow 0+}{longrightarrow} & f'(0) = -frac{5}{2}mbox{ for } f(t) = sqrt{1-5t +3t^2}
end{eqnarray*}$$
add a comment |
up vote
2
down vote
up vote
2
down vote
You may set $x = -frac{1}{t}$ and consider the limit for $stackrel{trightarrow 0+}{longrightarrow}$:
$$begin{eqnarray*} sqrt{x^2+5x+3}+x
& stackrel{x = -frac{1}{t}}{=} & frac{sqrt{1-5t +3t^2} - 1}{t} \
& stackrel{trightarrow 0+}{longrightarrow} & f'(0) = -frac{5}{2}mbox{ for } f(t) = sqrt{1-5t +3t^2}
end{eqnarray*}$$
You may set $x = -frac{1}{t}$ and consider the limit for $stackrel{trightarrow 0+}{longrightarrow}$:
$$begin{eqnarray*} sqrt{x^2+5x+3}+x
& stackrel{x = -frac{1}{t}}{=} & frac{sqrt{1-5t +3t^2} - 1}{t} \
& stackrel{trightarrow 0+}{longrightarrow} & f'(0) = -frac{5}{2}mbox{ for } f(t) = sqrt{1-5t +3t^2}
end{eqnarray*}$$
answered Dec 3 at 14:50
trancelocation
8,8951521
8,8951521
add a comment |
add a comment |
up vote
2
down vote
The first step is illegal, not the second. This is because $sqrt{x^2 + 5x + 3} notequiv -x$.
Also addressing your comment:
The idea behind $lim_{x to -infty} sqrt{x^2 + 5x + 3} = dots = lim_{x to -infty} −x$ was, that $x^2$ dominantes $5x+3$. I also thought if that argument is wrong here, if $lim_{x to -infty} x − x = 0$ is legal and everything of the same "type"; then I'm okay again. Then I have to think about why the domination argument doesn't work as I thought it does here.
You are right in thinking that $x^2$ dominates $5x+3$, so that $color{blue}{sqrt{x^2 + 5x + 3}} sim color{red}{-x}$ for large negative $x$. Indeed there is no problem when computing their ratio:
$$lim_{x to -infty} frac{color{blue}{sqrt{x^2 + 5x + 3}}}{color{red}{-x}} = 1$$
However, because you are computing the difference between $color{blue}{sqrt{x^2 + 5x + 3}}$ and $color{red}{-x}$, such an approximation is not fine enough. You need more terms of the series expansion:
$$
begin{align*}
color{blue}{sqrt{x^2 + 5x + 3}}
&= -x sqrt{1 + frac{5}{x} + frac{3}{x^2}} \
&sim -x left(1 + frac{1}{2} cdot frac{5}{x} + Oleft(frac{1}{x^2}right) right) \
&= color{red}{-x} - frac{5}{2} + Oleft(frac{1}{x}right)
end{align*}
$$
Then may you conclude that the required limit is $-5/2$.
By analogy, an approximation such as $color{blue}{x + 1} sim color{red}{x}$ is perfectly fine for computing the ratio limit $lim_{x to -infty} (color{blue}{x + 1}) / (color{red}{x})$. But if you are computing the difference limit, then you may not discard the $1$ in $color{blue}{x + 1}$.
add a comment |
up vote
2
down vote
The first step is illegal, not the second. This is because $sqrt{x^2 + 5x + 3} notequiv -x$.
Also addressing your comment:
The idea behind $lim_{x to -infty} sqrt{x^2 + 5x + 3} = dots = lim_{x to -infty} −x$ was, that $x^2$ dominantes $5x+3$. I also thought if that argument is wrong here, if $lim_{x to -infty} x − x = 0$ is legal and everything of the same "type"; then I'm okay again. Then I have to think about why the domination argument doesn't work as I thought it does here.
You are right in thinking that $x^2$ dominates $5x+3$, so that $color{blue}{sqrt{x^2 + 5x + 3}} sim color{red}{-x}$ for large negative $x$. Indeed there is no problem when computing their ratio:
$$lim_{x to -infty} frac{color{blue}{sqrt{x^2 + 5x + 3}}}{color{red}{-x}} = 1$$
However, because you are computing the difference between $color{blue}{sqrt{x^2 + 5x + 3}}$ and $color{red}{-x}$, such an approximation is not fine enough. You need more terms of the series expansion:
$$
begin{align*}
color{blue}{sqrt{x^2 + 5x + 3}}
&= -x sqrt{1 + frac{5}{x} + frac{3}{x^2}} \
&sim -x left(1 + frac{1}{2} cdot frac{5}{x} + Oleft(frac{1}{x^2}right) right) \
&= color{red}{-x} - frac{5}{2} + Oleft(frac{1}{x}right)
end{align*}
$$
Then may you conclude that the required limit is $-5/2$.
By analogy, an approximation such as $color{blue}{x + 1} sim color{red}{x}$ is perfectly fine for computing the ratio limit $lim_{x to -infty} (color{blue}{x + 1}) / (color{red}{x})$. But if you are computing the difference limit, then you may not discard the $1$ in $color{blue}{x + 1}$.
add a comment |
up vote
2
down vote
up vote
2
down vote
The first step is illegal, not the second. This is because $sqrt{x^2 + 5x + 3} notequiv -x$.
Also addressing your comment:
The idea behind $lim_{x to -infty} sqrt{x^2 + 5x + 3} = dots = lim_{x to -infty} −x$ was, that $x^2$ dominantes $5x+3$. I also thought if that argument is wrong here, if $lim_{x to -infty} x − x = 0$ is legal and everything of the same "type"; then I'm okay again. Then I have to think about why the domination argument doesn't work as I thought it does here.
You are right in thinking that $x^2$ dominates $5x+3$, so that $color{blue}{sqrt{x^2 + 5x + 3}} sim color{red}{-x}$ for large negative $x$. Indeed there is no problem when computing their ratio:
$$lim_{x to -infty} frac{color{blue}{sqrt{x^2 + 5x + 3}}}{color{red}{-x}} = 1$$
However, because you are computing the difference between $color{blue}{sqrt{x^2 + 5x + 3}}$ and $color{red}{-x}$, such an approximation is not fine enough. You need more terms of the series expansion:
$$
begin{align*}
color{blue}{sqrt{x^2 + 5x + 3}}
&= -x sqrt{1 + frac{5}{x} + frac{3}{x^2}} \
&sim -x left(1 + frac{1}{2} cdot frac{5}{x} + Oleft(frac{1}{x^2}right) right) \
&= color{red}{-x} - frac{5}{2} + Oleft(frac{1}{x}right)
end{align*}
$$
Then may you conclude that the required limit is $-5/2$.
By analogy, an approximation such as $color{blue}{x + 1} sim color{red}{x}$ is perfectly fine for computing the ratio limit $lim_{x to -infty} (color{blue}{x + 1}) / (color{red}{x})$. But if you are computing the difference limit, then you may not discard the $1$ in $color{blue}{x + 1}$.
The first step is illegal, not the second. This is because $sqrt{x^2 + 5x + 3} notequiv -x$.
Also addressing your comment:
The idea behind $lim_{x to -infty} sqrt{x^2 + 5x + 3} = dots = lim_{x to -infty} −x$ was, that $x^2$ dominantes $5x+3$. I also thought if that argument is wrong here, if $lim_{x to -infty} x − x = 0$ is legal and everything of the same "type"; then I'm okay again. Then I have to think about why the domination argument doesn't work as I thought it does here.
You are right in thinking that $x^2$ dominates $5x+3$, so that $color{blue}{sqrt{x^2 + 5x + 3}} sim color{red}{-x}$ for large negative $x$. Indeed there is no problem when computing their ratio:
$$lim_{x to -infty} frac{color{blue}{sqrt{x^2 + 5x + 3}}}{color{red}{-x}} = 1$$
However, because you are computing the difference between $color{blue}{sqrt{x^2 + 5x + 3}}$ and $color{red}{-x}$, such an approximation is not fine enough. You need more terms of the series expansion:
$$
begin{align*}
color{blue}{sqrt{x^2 + 5x + 3}}
&= -x sqrt{1 + frac{5}{x} + frac{3}{x^2}} \
&sim -x left(1 + frac{1}{2} cdot frac{5}{x} + Oleft(frac{1}{x^2}right) right) \
&= color{red}{-x} - frac{5}{2} + Oleft(frac{1}{x}right)
end{align*}
$$
Then may you conclude that the required limit is $-5/2$.
By analogy, an approximation such as $color{blue}{x + 1} sim color{red}{x}$ is perfectly fine for computing the ratio limit $lim_{x to -infty} (color{blue}{x + 1}) / (color{red}{x})$. But if you are computing the difference limit, then you may not discard the $1$ in $color{blue}{x + 1}$.
edited Dec 4 at 6:43
answered Dec 4 at 6:24
lastresort
1,295414
1,295414
add a comment |
add a comment |
up vote
1
down vote
Possibly the biggest problem in your first paragraph is that $sqrt{text{whatever}}$ is non-negative when we are working over the reals. Can you exhibit any choice of $x$ in the real numbers giving a negative value of $sqrt{x^2 + 5 x + 3}$?
add a comment |
up vote
1
down vote
Possibly the biggest problem in your first paragraph is that $sqrt{text{whatever}}$ is non-negative when we are working over the reals. Can you exhibit any choice of $x$ in the real numbers giving a negative value of $sqrt{x^2 + 5 x + 3}$?
add a comment |
up vote
1
down vote
up vote
1
down vote
Possibly the biggest problem in your first paragraph is that $sqrt{text{whatever}}$ is non-negative when we are working over the reals. Can you exhibit any choice of $x$ in the real numbers giving a negative value of $sqrt{x^2 + 5 x + 3}$?
Possibly the biggest problem in your first paragraph is that $sqrt{text{whatever}}$ is non-negative when we are working over the reals. Can you exhibit any choice of $x$ in the real numbers giving a negative value of $sqrt{x^2 + 5 x + 3}$?
answered Dec 3 at 20:33
Eric Towers
31.5k22265
31.5k22265
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up vote
0
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I will flesh out much of what would be considered a series of valid steps you may take in order to get something similar to the first step performed in the initial limit calculation. I will note that the second step that was taken is valid because we perform all of the algebra in the limit expression first, then we take a limit of what results.
The following line of reasoning shows that whenever you have $sqrt{x^2+beta x+alpha}$ as a term (added or substracted, not multiplied or divided) in a limit, it may be replaced freely by $|x+frac{beta}{2}|$ inside the limit (of course slight adjustments must be made for it to be shown in general).
Note that for every $b>frac{5}{2}$,
$$lim_{xto-infty}sqrt{x^2+2bx+b^2}+x lelim_{xto-infty}sqrt{x^2+5x+3}+x.$$
To see why this is the case, here is a series of inequalities. First pick a $b>frac{5}{2}$. Then pick $x$ so that $xlefrac{3-b^2}{2b-5}$.
Then
$$(2b-5)xle 3-b^2,$$ $$2bx+b^2le 5x+3,$$
$$x^2+2bx+b^2le x^2+5x+3,$$
$$sqrt{x^2+2bx+b^2}lesqrt{x^2+5x+3},$$
$$sqrt{x^2+2bx+b^2}+xlesqrt{x^2+5x+3}+x,$$
$$lim_{xto-infty}sqrt{x^2+2bx+b^2}+x lelim_{xto-infty}sqrt{x^2+5x+3}+x.$$
The intuition for this comes largely from working backward. So thus the claim is justified.
So now we can say that
$$lim_{bto frac{5}{2}^-}lim_{xto-infty}sqrt{x^2+2bx+b^2}+x lelim_{xto-infty}sqrt{x^2+5x+3}+x,$$
$$lim_{bto frac{5}{2}^-}|x+b|+x=lim_{bto frac{5}{2}^-}-x-b+x=lim_{bto frac{5}{2}^-}-b=-frac{5}{2}lelim_{xto-infty}sqrt{x^2+5x+3}+x.$$
Now since $3<frac{25}{4}$, we have that.
$$lim_{xto-infty}sqrt{x^2+5x+3}+xle lim_{xto-infty}sqrt{x^2+5x+frac{25}{4}}+x=lim_{xto-infty}|x+frac{5}{2}|+x=lim_{xto-infty}-x-frac{5}{2}+x=lim_{xto-infty}-frac{5}{2}=-frac{5}{2}.$$
So $$lim_{xto-infty}sqrt{x^2+5x+3}+x=-frac{5}{2}.$$
add a comment |
up vote
0
down vote
I will flesh out much of what would be considered a series of valid steps you may take in order to get something similar to the first step performed in the initial limit calculation. I will note that the second step that was taken is valid because we perform all of the algebra in the limit expression first, then we take a limit of what results.
The following line of reasoning shows that whenever you have $sqrt{x^2+beta x+alpha}$ as a term (added or substracted, not multiplied or divided) in a limit, it may be replaced freely by $|x+frac{beta}{2}|$ inside the limit (of course slight adjustments must be made for it to be shown in general).
Note that for every $b>frac{5}{2}$,
$$lim_{xto-infty}sqrt{x^2+2bx+b^2}+x lelim_{xto-infty}sqrt{x^2+5x+3}+x.$$
To see why this is the case, here is a series of inequalities. First pick a $b>frac{5}{2}$. Then pick $x$ so that $xlefrac{3-b^2}{2b-5}$.
Then
$$(2b-5)xle 3-b^2,$$ $$2bx+b^2le 5x+3,$$
$$x^2+2bx+b^2le x^2+5x+3,$$
$$sqrt{x^2+2bx+b^2}lesqrt{x^2+5x+3},$$
$$sqrt{x^2+2bx+b^2}+xlesqrt{x^2+5x+3}+x,$$
$$lim_{xto-infty}sqrt{x^2+2bx+b^2}+x lelim_{xto-infty}sqrt{x^2+5x+3}+x.$$
The intuition for this comes largely from working backward. So thus the claim is justified.
So now we can say that
$$lim_{bto frac{5}{2}^-}lim_{xto-infty}sqrt{x^2+2bx+b^2}+x lelim_{xto-infty}sqrt{x^2+5x+3}+x,$$
$$lim_{bto frac{5}{2}^-}|x+b|+x=lim_{bto frac{5}{2}^-}-x-b+x=lim_{bto frac{5}{2}^-}-b=-frac{5}{2}lelim_{xto-infty}sqrt{x^2+5x+3}+x.$$
Now since $3<frac{25}{4}$, we have that.
$$lim_{xto-infty}sqrt{x^2+5x+3}+xle lim_{xto-infty}sqrt{x^2+5x+frac{25}{4}}+x=lim_{xto-infty}|x+frac{5}{2}|+x=lim_{xto-infty}-x-frac{5}{2}+x=lim_{xto-infty}-frac{5}{2}=-frac{5}{2}.$$
So $$lim_{xto-infty}sqrt{x^2+5x+3}+x=-frac{5}{2}.$$
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I will flesh out much of what would be considered a series of valid steps you may take in order to get something similar to the first step performed in the initial limit calculation. I will note that the second step that was taken is valid because we perform all of the algebra in the limit expression first, then we take a limit of what results.
The following line of reasoning shows that whenever you have $sqrt{x^2+beta x+alpha}$ as a term (added or substracted, not multiplied or divided) in a limit, it may be replaced freely by $|x+frac{beta}{2}|$ inside the limit (of course slight adjustments must be made for it to be shown in general).
Note that for every $b>frac{5}{2}$,
$$lim_{xto-infty}sqrt{x^2+2bx+b^2}+x lelim_{xto-infty}sqrt{x^2+5x+3}+x.$$
To see why this is the case, here is a series of inequalities. First pick a $b>frac{5}{2}$. Then pick $x$ so that $xlefrac{3-b^2}{2b-5}$.
Then
$$(2b-5)xle 3-b^2,$$ $$2bx+b^2le 5x+3,$$
$$x^2+2bx+b^2le x^2+5x+3,$$
$$sqrt{x^2+2bx+b^2}lesqrt{x^2+5x+3},$$
$$sqrt{x^2+2bx+b^2}+xlesqrt{x^2+5x+3}+x,$$
$$lim_{xto-infty}sqrt{x^2+2bx+b^2}+x lelim_{xto-infty}sqrt{x^2+5x+3}+x.$$
The intuition for this comes largely from working backward. So thus the claim is justified.
So now we can say that
$$lim_{bto frac{5}{2}^-}lim_{xto-infty}sqrt{x^2+2bx+b^2}+x lelim_{xto-infty}sqrt{x^2+5x+3}+x,$$
$$lim_{bto frac{5}{2}^-}|x+b|+x=lim_{bto frac{5}{2}^-}-x-b+x=lim_{bto frac{5}{2}^-}-b=-frac{5}{2}lelim_{xto-infty}sqrt{x^2+5x+3}+x.$$
Now since $3<frac{25}{4}$, we have that.
$$lim_{xto-infty}sqrt{x^2+5x+3}+xle lim_{xto-infty}sqrt{x^2+5x+frac{25}{4}}+x=lim_{xto-infty}|x+frac{5}{2}|+x=lim_{xto-infty}-x-frac{5}{2}+x=lim_{xto-infty}-frac{5}{2}=-frac{5}{2}.$$
So $$lim_{xto-infty}sqrt{x^2+5x+3}+x=-frac{5}{2}.$$
I will flesh out much of what would be considered a series of valid steps you may take in order to get something similar to the first step performed in the initial limit calculation. I will note that the second step that was taken is valid because we perform all of the algebra in the limit expression first, then we take a limit of what results.
The following line of reasoning shows that whenever you have $sqrt{x^2+beta x+alpha}$ as a term (added or substracted, not multiplied or divided) in a limit, it may be replaced freely by $|x+frac{beta}{2}|$ inside the limit (of course slight adjustments must be made for it to be shown in general).
Note that for every $b>frac{5}{2}$,
$$lim_{xto-infty}sqrt{x^2+2bx+b^2}+x lelim_{xto-infty}sqrt{x^2+5x+3}+x.$$
To see why this is the case, here is a series of inequalities. First pick a $b>frac{5}{2}$. Then pick $x$ so that $xlefrac{3-b^2}{2b-5}$.
Then
$$(2b-5)xle 3-b^2,$$ $$2bx+b^2le 5x+3,$$
$$x^2+2bx+b^2le x^2+5x+3,$$
$$sqrt{x^2+2bx+b^2}lesqrt{x^2+5x+3},$$
$$sqrt{x^2+2bx+b^2}+xlesqrt{x^2+5x+3}+x,$$
$$lim_{xto-infty}sqrt{x^2+2bx+b^2}+x lelim_{xto-infty}sqrt{x^2+5x+3}+x.$$
The intuition for this comes largely from working backward. So thus the claim is justified.
So now we can say that
$$lim_{bto frac{5}{2}^-}lim_{xto-infty}sqrt{x^2+2bx+b^2}+x lelim_{xto-infty}sqrt{x^2+5x+3}+x,$$
$$lim_{bto frac{5}{2}^-}|x+b|+x=lim_{bto frac{5}{2}^-}-x-b+x=lim_{bto frac{5}{2}^-}-b=-frac{5}{2}lelim_{xto-infty}sqrt{x^2+5x+3}+x.$$
Now since $3<frac{25}{4}$, we have that.
$$lim_{xto-infty}sqrt{x^2+5x+3}+xle lim_{xto-infty}sqrt{x^2+5x+frac{25}{4}}+x=lim_{xto-infty}|x+frac{5}{2}|+x=lim_{xto-infty}-x-frac{5}{2}+x=lim_{xto-infty}-frac{5}{2}=-frac{5}{2}.$$
So $$lim_{xto-infty}sqrt{x^2+5x+3}+x=-frac{5}{2}.$$
edited Dec 5 at 2:48
answered Dec 4 at 6:22
John B
1766
1766
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What's the issue with $x^2-x^2=0$? It's a perfectly legal statement. One shouldn't doubt oneself too much. On the other hand the issue with your first attempt is that you can't replace $sqrt{x^2+5x+3}$ with $-x$ as they are not equal. One can only replace one expression with another expression when both are equal.
– Paramanand Singh
Dec 4 at 3:57
Technically, your first two inequalities are not wrong because all terms equal 0.
– Dirk
Dec 4 at 5:55
How did you even make that huge leap of faith to go from $sqrt{x^2+5x+3}$ to $-x$? Even if you made $sqrt{x^2+5x+3}$ become $x$, although wrong, this would still be understandable, but where did the $-$ pop up from?
– YiFan
Dec 5 at 2:55
@YiFan: you should note that $x$ is negative and square root is always non-negative so $-x$ is a reasonable choice.
– Paramanand Singh
Dec 5 at 3:34