To prove that the limit of a bi-variate function is nonexistent at a point [duplicate]
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This question already has an answer here:
Non-existence of $lim limits_{(x, y) to (0,0)} frac{x^3 + y^3}{x - y} $
3 answers
It has been asked to evaluate $$lim_{(x,y)to(0,0)} frac{x^3 + y^3}{x-y}$$ if it exists at all and otherwise to disprove it.
After much thoughts , I came up with an idea of substituting $y$ with $x-mx^3$ which devolved the limit to $(2/m)$ and thus served my purpose of disproving the existence of a limit.
But, thinking of such half-weird substitutions take some reasonable amount of time; the luxury of which is seldom available at examinations. So, what are other better methods to disprove the existence of this limit?
Any general algorithm (??) on disproving the existence of bi-variate limits (without indulging into trick-substitutions) will be also appreciated.
real-analysis limits multivariable-calculus
edited Dec 16 '18 at 19:01
Rebellos
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asked Dec 16 '18 at 18:57
Winged Blades of GodricWinged Blades of Godric
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marked as duplicate by Saad, RRL real-analysis
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Dec 18 '18 at 2:09
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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show 4 more comments
$begingroup$
This question already has an answer here:
Non-existence of $lim limits_{(x, y) to (0,0)} frac{x^3 + y^3}{x - y} $
3 answers
It has been asked to evaluate $$lim_{(x,y)to(0,0)} frac{x^3 + y^3}{x-y}$$ if it exists at all and otherwise to disprove it.
After much thoughts , I came up with an idea of substituting $y$ with $x-mx^3$ which devolved the limit to $(2/m)$ and thus served my purpose of disproving the existence of a limit.
But, thinking of such half-weird substitutions take some reasonable amount of time; the luxury of which is seldom available at examinations. So, what are other better methods to disprove the existence of this limit?
Any general algorithm (??) on disproving the existence of bi-variate limits (without indulging into trick-substitutions) will be also appreciated.
real-analysis limits multivariable-calculus
edited Dec 16 '18 at 19:01
Rebellos
15.3k31250
asked Dec 16 '18 at 18:57
Winged Blades of GodricWinged Blades of Godric
665
$endgroup$
marked as duplicate by Saad, RRL real-analysis
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Dec 18 '18 at 2:09
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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@M.Santos Well, in that case, I need to show the trig function to be unbounded.....Seems tedious and complex:(...... (In response to a now-deleted comment, that mentioned about transformation to polar coordinates.)
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– Winged Blades of Godric
Dec 16 '18 at 19:11
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What about $x=y$?
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– Robert Israel
Dec 16 '18 at 19:22
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@RobertIsrael presumably the domain of the function which is having its limit taken does not include points in $mathbb{R}^2$ of the form $(x,x)$.
$endgroup$
– E-mu
Dec 16 '18 at 19:30
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@Robert Apologies. The domain excludes all points in the form of $(x,x)$.
$endgroup$
– Winged Blades of Godric
Dec 16 '18 at 19:32
1
$begingroup$
Then take $x$ very close to $y$.
$endgroup$
– Robert Israel
Dec 16 '18 at 19:40
|
show 4 more comments
$begingroup$
This question already has an answer here:
Non-existence of $lim limits_{(x, y) to (0,0)} frac{x^3 + y^3}{x - y} $
3 answers
It has been asked to evaluate $$lim_{(x,y)to(0,0)} frac{x^3 + y^3}{x-y}$$ if it exists at all and otherwise to disprove it.
After much thoughts , I came up with an idea of substituting $y$ with $x-mx^3$ which devolved the limit to $(2/m)$ and thus served my purpose of disproving the existence of a limit.
But, thinking of such half-weird substitutions take some reasonable amount of time; the luxury of which is seldom available at examinations. So, what are other better methods to disprove the existence of this limit?
Any general algorithm (??) on disproving the existence of bi-variate limits (without indulging into trick-substitutions) will be also appreciated.
real-analysis limits multivariable-calculus
edited Dec 16 '18 at 19:01
Rebellos
15.3k31250
asked Dec 16 '18 at 18:57
Winged Blades of GodricWinged Blades of Godric
665
$endgroup$
This question already has an answer here:
Non-existence of $lim limits_{(x, y) to (0,0)} frac{x^3 + y^3}{x - y} $
3 answers
It has been asked to evaluate $$lim_{(x,y)to(0,0)} frac{x^3 + y^3}{x-y}$$ if it exists at all and otherwise to disprove it.
After much thoughts , I came up with an idea of substituting $y$ with $x-mx^3$ which devolved the limit to $(2/m)$ and thus served my purpose of disproving the existence of a limit.
But, thinking of such half-weird substitutions take some reasonable amount of time; the luxury of which is seldom available at examinations. So, what are other better methods to disprove the existence of this limit?
Any general algorithm (??) on disproving the existence of bi-variate limits (without indulging into trick-substitutions) will be also appreciated.
This question already has an answer here:
Non-existence of $lim limits_{(x, y) to (0,0)} frac{x^3 + y^3}{x - y} $
3 answers
real-analysis limits multivariable-calculus
real-analysis limits multivariable-calculus
edited Dec 16 '18 at 19:01
Rebellos
15.3k31250
asked Dec 16 '18 at 18:57
Winged Blades of GodricWinged Blades of Godric
665
edited Dec 16 '18 at 19:01
Rebellos
15.3k31250
asked Dec 16 '18 at 18:57
Winged Blades of GodricWinged Blades of Godric
665
edited Dec 16 '18 at 19:01
Rebellos
15.3k31250
edited Dec 16 '18 at 19:01
Rebellos
15.3k31250
edited Dec 16 '18 at 19:01
Rebellos
15.3k31250
15.3k31250
asked Dec 16 '18 at 18:57
Winged Blades of GodricWinged Blades of Godric
665
asked Dec 16 '18 at 18:57
Winged Blades of GodricWinged Blades of Godric
665
asked Dec 16 '18 at 18:57
Winged Blades of GodricWinged Blades of Godric
665
665
marked as duplicate by Saad, RRL real-analysis
Users with the real-analysis badge can single-handedly close real-analysis questions as duplicates and reopen them as needed.
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Dec 18 '18 at 2:09
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Saad, RRL real-analysis
Users with the real-analysis badge can single-handedly close real-analysis questions as duplicates and reopen them as needed.
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Dec 18 '18 at 2:09
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
$begingroup$
@M.Santos Well, in that case, I need to show the trig function to be unbounded.....Seems tedious and complex:(...... (In response to a now-deleted comment, that mentioned about transformation to polar coordinates.)
$endgroup$
– Winged Blades of Godric
Dec 16 '18 at 19:11
$begingroup$
What about $x=y$?
$endgroup$
– Robert Israel
Dec 16 '18 at 19:22
$begingroup$
@RobertIsrael presumably the domain of the function which is having its limit taken does not include points in $mathbb{R}^2$ of the form $(x,x)$.
$endgroup$
– E-mu
Dec 16 '18 at 19:30
$begingroup$
@Robert Apologies. The domain excludes all points in the form of $(x,x)$.
$endgroup$
– Winged Blades of Godric
Dec 16 '18 at 19:32
1
$begingroup$
Then take $x$ very close to $y$.
$endgroup$
– Robert Israel
Dec 16 '18 at 19:40
|
show 4 more comments
$begingroup$
@M.Santos Well, in that case, I need to show the trig function to be unbounded.....Seems tedious and complex:(...... (In response to a now-deleted comment, that mentioned about transformation to polar coordinates.)
$endgroup$
– Winged Blades of Godric
Dec 16 '18 at 19:11
$begingroup$
What about $x=y$?
$endgroup$
– Robert Israel
Dec 16 '18 at 19:22
$begingroup$
@RobertIsrael presumably the domain of the function which is having its limit taken does not include points in $mathbb{R}^2$ of the form $(x,x)$.
$endgroup$
– E-mu
Dec 16 '18 at 19:30
$begingroup$
@Robert Apologies. The domain excludes all points in the form of $(x,x)$.
$endgroup$
– Winged Blades of Godric
Dec 16 '18 at 19:32
1
$begingroup$
Then take $x$ very close to $y$.
$endgroup$
– Robert Israel
Dec 16 '18 at 19:40
$begingroup$
@M.Santos Well, in that case, I need to show the trig function to be unbounded.....Seems tedious and complex:(...... (In response to a now-deleted comment, that mentioned about transformation to polar coordinates.)
$endgroup$
– Winged Blades of Godric
Dec 16 '18 at 19:11
$begingroup$
@M.Santos Well, in that case, I need to show the trig function to be unbounded.....Seems tedious and complex:(...... (In response to a now-deleted comment, that mentioned about transformation to polar coordinates.)
$endgroup$
– Winged Blades of Godric
Dec 16 '18 at 19:11
$begingroup$
What about $x=y$?
$endgroup$
– Robert Israel
Dec 16 '18 at 19:22
$begingroup$
What about $x=y$?
$endgroup$
– Robert Israel
Dec 16 '18 at 19:22
$begingroup$
@RobertIsrael presumably the domain of the function which is having its limit taken does not include points in $mathbb{R}^2$ of the form $(x,x)$.
$endgroup$
– E-mu
Dec 16 '18 at 19:30
$begingroup$
@RobertIsrael presumably the domain of the function which is having its limit taken does not include points in $mathbb{R}^2$ of the form $(x,x)$.
$endgroup$
– E-mu
Dec 16 '18 at 19:30
$begingroup$
@Robert Apologies. The domain excludes all points in the form of $(x,x)$.
$endgroup$
– Winged Blades of Godric
Dec 16 '18 at 19:32
$begingroup$
@Robert Apologies. The domain excludes all points in the form of $(x,x)$.
$endgroup$
– Winged Blades of Godric
Dec 16 '18 at 19:32
1
1
$begingroup$
Then take $x$ very close to $y$.
$endgroup$
– Robert Israel
Dec 16 '18 at 19:40
$begingroup$
Then take $x$ very close to $y$.
$endgroup$
– Robert Israel
Dec 16 '18 at 19:40
|
show 4 more comments
1 Answer
1
active
oldest
votes
$begingroup$
The standard and more effective method to show (by hand calculation) that a limit doesn't exist is to find at least two different paths with different limits as you have done.
Of course we can apply the epsilon-delta method using the same two paths but that way shouldn't be so different form the method you have already used to prove that limit doesn't exist.
Note that with some practice and a good strategy, figure out such "half-weird substitutions" don't take much time and it is indeed a correct an effective method to proceed. For the proper strategy to follow, refer also to the related
- What is $lim_{(x,y)to(0,0)}frac{ (x^2y^2}{(x^3-y^3)}$?
answered Dec 17 '18 at 17:35
gimusigimusi
93k84594
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The standard and more effective method to show (by hand calculation) that a limit doesn't exist is to find at least two different paths with different limits as you have done.
Of course we can apply the epsilon-delta method using the same two paths but that way shouldn't be so different form the method you have already used to prove that limit doesn't exist.
Note that with some practice and a good strategy, figure out such "half-weird substitutions" don't take much time and it is indeed a correct an effective method to proceed. For the proper strategy to follow, refer also to the related
- What is $lim_{(x,y)to(0,0)}frac{ (x^2y^2}{(x^3-y^3)}$?
answered Dec 17 '18 at 17:35
gimusigimusi
93k84594
$endgroup$
add a comment |
$begingroup$
The standard and more effective method to show (by hand calculation) that a limit doesn't exist is to find at least two different paths with different limits as you have done.
Of course we can apply the epsilon-delta method using the same two paths but that way shouldn't be so different form the method you have already used to prove that limit doesn't exist.
Note that with some practice and a good strategy, figure out such "half-weird substitutions" don't take much time and it is indeed a correct an effective method to proceed. For the proper strategy to follow, refer also to the related
- What is $lim_{(x,y)to(0,0)}frac{ (x^2y^2}{(x^3-y^3)}$?
answered Dec 17 '18 at 17:35
gimusigimusi
93k84594
$endgroup$
add a comment |
$begingroup$
The standard and more effective method to show (by hand calculation) that a limit doesn't exist is to find at least two different paths with different limits as you have done.
Of course we can apply the epsilon-delta method using the same two paths but that way shouldn't be so different form the method you have already used to prove that limit doesn't exist.
Note that with some practice and a good strategy, figure out such "half-weird substitutions" don't take much time and it is indeed a correct an effective method to proceed. For the proper strategy to follow, refer also to the related
- What is $lim_{(x,y)to(0,0)}frac{ (x^2y^2}{(x^3-y^3)}$?
answered Dec 17 '18 at 17:35
gimusigimusi
93k84594
$endgroup$
The standard and more effective method to show (by hand calculation) that a limit doesn't exist is to find at least two different paths with different limits as you have done.
Of course we can apply the epsilon-delta method using the same two paths but that way shouldn't be so different form the method you have already used to prove that limit doesn't exist.
Note that with some practice and a good strategy, figure out such "half-weird substitutions" don't take much time and it is indeed a correct an effective method to proceed. For the proper strategy to follow, refer also to the related
- What is $lim_{(x,y)to(0,0)}frac{ (x^2y^2}{(x^3-y^3)}$?
answered Dec 17 '18 at 17:35
gimusigimusi
93k84594
answered Dec 17 '18 at 17:35
gimusigimusi
93k84594
answered Dec 17 '18 at 17:35
gimusigimusi
93k84594
answered Dec 17 '18 at 17:35
gimusigimusi
93k84594
93k84594
add a comment |
add a comment |
$begingroup$
@M.Santos Well, in that case, I need to show the trig function to be unbounded.....Seems tedious and complex:(...... (In response to a now-deleted comment, that mentioned about transformation to polar coordinates.)
$endgroup$
– Winged Blades of Godric
Dec 16 '18 at 19:11
$begingroup$
What about $x=y$?
$endgroup$
– Robert Israel
Dec 16 '18 at 19:22
$begingroup$
@RobertIsrael presumably the domain of the function which is having its limit taken does not include points in $mathbb{R}^2$ of the form $(x,x)$.
$endgroup$
– E-mu
Dec 16 '18 at 19:30
$begingroup$
@Robert Apologies. The domain excludes all points in the form of $(x,x)$.
$endgroup$
– Winged Blades of Godric
Dec 16 '18 at 19:32
1
$begingroup$
Then take $x$ very close to $y$.
$endgroup$
– Robert Israel
Dec 16 '18 at 19:40