When do the limit sum and product rules not apply. [closed]
$begingroup$
For f and g defined on all real numbers, when is
$$lim_{xto a} (f(x) + g(x)) ne lim_{xto a} f(x) + lim_{xto a} g(x)$$
and when is
$$lim_{xto a} (f(x) cdot g(x)) ne lim_{xto a} f(x) cdot lim_{xto a} g(x)$$
calculus limits
edited Dec 2 '18 at 1:03
Larry
2,0592826
asked Dec 2 '18 at 0:26
HarshitHarshit
11
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closed as off-topic by Saad, KReiser, qbert, Jyrki Lahtonen, NCh Dec 3 '18 at 4:59
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Saad, KReiser, qbert, Jyrki Lahtonen, NCh
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
For f and g defined on all real numbers, when is
$$lim_{xto a} (f(x) + g(x)) ne lim_{xto a} f(x) + lim_{xto a} g(x)$$
and when is
$$lim_{xto a} (f(x) cdot g(x)) ne lim_{xto a} f(x) cdot lim_{xto a} g(x)$$
calculus limits
edited Dec 2 '18 at 1:03
Larry
2,0592826
asked Dec 2 '18 at 0:26
HarshitHarshit
11
$endgroup$
closed as off-topic by Saad, KReiser, qbert, Jyrki Lahtonen, NCh Dec 3 '18 at 4:59
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Saad, KReiser, qbert, Jyrki Lahtonen, NCh
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
For f and g defined on all real numbers, when is
$$lim_{xto a} (f(x) + g(x)) ne lim_{xto a} f(x) + lim_{xto a} g(x)$$
and when is
$$lim_{xto a} (f(x) cdot g(x)) ne lim_{xto a} f(x) cdot lim_{xto a} g(x)$$
calculus limits
edited Dec 2 '18 at 1:03
Larry
2,0592826
asked Dec 2 '18 at 0:26
HarshitHarshit
11
$endgroup$
For f and g defined on all real numbers, when is
$$lim_{xto a} (f(x) + g(x)) ne lim_{xto a} f(x) + lim_{xto a} g(x)$$
and when is
$$lim_{xto a} (f(x) cdot g(x)) ne lim_{xto a} f(x) cdot lim_{xto a} g(x)$$
calculus limits
calculus limits
edited Dec 2 '18 at 1:03
Larry
2,0592826
asked Dec 2 '18 at 0:26
HarshitHarshit
11
edited Dec 2 '18 at 1:03
Larry
2,0592826
asked Dec 2 '18 at 0:26
HarshitHarshit
11
edited Dec 2 '18 at 1:03
Larry
2,0592826
edited Dec 2 '18 at 1:03
Larry
2,0592826
edited Dec 2 '18 at 1:03
Larry
2,0592826
2,0592826
asked Dec 2 '18 at 0:26
HarshitHarshit
11
asked Dec 2 '18 at 0:26
HarshitHarshit
11
asked Dec 2 '18 at 0:26
HarshitHarshit
11
11
closed as off-topic by Saad, KReiser, qbert, Jyrki Lahtonen, NCh Dec 3 '18 at 4:59
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Saad, KReiser, qbert, Jyrki Lahtonen, NCh
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Saad, KReiser, qbert, Jyrki Lahtonen, NCh Dec 3 '18 at 4:59
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Saad, KReiser, qbert, Jyrki Lahtonen, NCh
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Depends what you have as a range, but if it's also the real numbers, then it is always true that when the limits for $f$ and $g$ exist,
$$
lim_{xrightarrow a}(f(x)+g(x))=lim_{xrightarrow a}f(x)+lim_{xrightarrow a}g(x)
$$
and
$$
lim_{xrightarrow a}(f(x)*g(x))=lim_{xrightarrow a}f(x)*lim_{xrightarrow a}g(x)
$$
answered Dec 2 '18 at 0:44
NL1992NL1992
8311
$endgroup$
1
$begingroup$
This isn't really a good answer, I'm afraid. There are plenty of examples where $lim_{x to a} (f(x) + g(x))$ exists, but the other two fail to exist.
$endgroup$
– T. Bongers
Dec 2 '18 at 1:09
$begingroup$
But are there any examples where all three exist but they are not equal?
$endgroup$
– Harshit
Dec 2 '18 at 1:14
$begingroup$
You are right, I stand corrected. when the limits exist, though, this is true and you can easily find the proof in any elementary book on Calculus.
$endgroup$
– NL1992
Dec 2 '18 at 1:32
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Depends what you have as a range, but if it's also the real numbers, then it is always true that when the limits for $f$ and $g$ exist,
$$
lim_{xrightarrow a}(f(x)+g(x))=lim_{xrightarrow a}f(x)+lim_{xrightarrow a}g(x)
$$
and
$$
lim_{xrightarrow a}(f(x)*g(x))=lim_{xrightarrow a}f(x)*lim_{xrightarrow a}g(x)
$$
answered Dec 2 '18 at 0:44
NL1992NL1992
8311
$endgroup$
1
$begingroup$
This isn't really a good answer, I'm afraid. There are plenty of examples where $lim_{x to a} (f(x) + g(x))$ exists, but the other two fail to exist.
$endgroup$
– T. Bongers
Dec 2 '18 at 1:09
$begingroup$
But are there any examples where all three exist but they are not equal?
$endgroup$
– Harshit
Dec 2 '18 at 1:14
$begingroup$
You are right, I stand corrected. when the limits exist, though, this is true and you can easily find the proof in any elementary book on Calculus.
$endgroup$
– NL1992
Dec 2 '18 at 1:32
add a comment |
$begingroup$
Depends what you have as a range, but if it's also the real numbers, then it is always true that when the limits for $f$ and $g$ exist,
$$
lim_{xrightarrow a}(f(x)+g(x))=lim_{xrightarrow a}f(x)+lim_{xrightarrow a}g(x)
$$
and
$$
lim_{xrightarrow a}(f(x)*g(x))=lim_{xrightarrow a}f(x)*lim_{xrightarrow a}g(x)
$$
answered Dec 2 '18 at 0:44
NL1992NL1992
8311
$endgroup$
1
$begingroup$
This isn't really a good answer, I'm afraid. There are plenty of examples where $lim_{x to a} (f(x) + g(x))$ exists, but the other two fail to exist.
$endgroup$
– T. Bongers
Dec 2 '18 at 1:09
$begingroup$
But are there any examples where all three exist but they are not equal?
$endgroup$
– Harshit
Dec 2 '18 at 1:14
$begingroup$
You are right, I stand corrected. when the limits exist, though, this is true and you can easily find the proof in any elementary book on Calculus.
$endgroup$
– NL1992
Dec 2 '18 at 1:32
add a comment |
$begingroup$
Depends what you have as a range, but if it's also the real numbers, then it is always true that when the limits for $f$ and $g$ exist,
$$
lim_{xrightarrow a}(f(x)+g(x))=lim_{xrightarrow a}f(x)+lim_{xrightarrow a}g(x)
$$
and
$$
lim_{xrightarrow a}(f(x)*g(x))=lim_{xrightarrow a}f(x)*lim_{xrightarrow a}g(x)
$$
answered Dec 2 '18 at 0:44
NL1992NL1992
8311
$endgroup$
Depends what you have as a range, but if it's also the real numbers, then it is always true that when the limits for $f$ and $g$ exist,
$$
lim_{xrightarrow a}(f(x)+g(x))=lim_{xrightarrow a}f(x)+lim_{xrightarrow a}g(x)
$$
and
$$
lim_{xrightarrow a}(f(x)*g(x))=lim_{xrightarrow a}f(x)*lim_{xrightarrow a}g(x)
$$
answered Dec 2 '18 at 0:44
NL1992NL1992
8311
edited Dec 2 '18 at 1:31
answered Dec 2 '18 at 0:44
NL1992NL1992
8311
answered Dec 2 '18 at 0:44
NL1992NL1992
8311
answered Dec 2 '18 at 0:44
NL1992NL1992
8311
8311
1
$begingroup$
This isn't really a good answer, I'm afraid. There are plenty of examples where $lim_{x to a} (f(x) + g(x))$ exists, but the other two fail to exist.
$endgroup$
– T. Bongers
Dec 2 '18 at 1:09
$begingroup$
But are there any examples where all three exist but they are not equal?
$endgroup$
– Harshit
Dec 2 '18 at 1:14
$begingroup$
You are right, I stand corrected. when the limits exist, though, this is true and you can easily find the proof in any elementary book on Calculus.
$endgroup$
– NL1992
Dec 2 '18 at 1:32
add a comment |
1
$begingroup$
This isn't really a good answer, I'm afraid. There are plenty of examples where $lim_{x to a} (f(x) + g(x))$ exists, but the other two fail to exist.
$endgroup$
– T. Bongers
Dec 2 '18 at 1:09
$begingroup$
But are there any examples where all three exist but they are not equal?
$endgroup$
– Harshit
Dec 2 '18 at 1:14
$begingroup$
You are right, I stand corrected. when the limits exist, though, this is true and you can easily find the proof in any elementary book on Calculus.
$endgroup$
– NL1992
Dec 2 '18 at 1:32
1
1
$begingroup$
This isn't really a good answer, I'm afraid. There are plenty of examples where $lim_{x to a} (f(x) + g(x))$ exists, but the other two fail to exist.
$endgroup$
– T. Bongers
Dec 2 '18 at 1:09
$begingroup$
This isn't really a good answer, I'm afraid. There are plenty of examples where $lim_{x to a} (f(x) + g(x))$ exists, but the other two fail to exist.
$endgroup$
– T. Bongers
Dec 2 '18 at 1:09
$begingroup$
But are there any examples where all three exist but they are not equal?
$endgroup$
– Harshit
Dec 2 '18 at 1:14
$begingroup$
But are there any examples where all three exist but they are not equal?
$endgroup$
– Harshit
Dec 2 '18 at 1:14
$begingroup$
You are right, I stand corrected. when the limits exist, though, this is true and you can easily find the proof in any elementary book on Calculus.
$endgroup$
– NL1992
Dec 2 '18 at 1:32
$begingroup$
You are right, I stand corrected. when the limits exist, though, this is true and you can easily find the proof in any elementary book on Calculus.
$endgroup$
– NL1992
Dec 2 '18 at 1:32
add a comment |
