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Multivariate limit of $|x|^y$

How do I find the limit of $fracxysqrtx^2 + xy + y^2$ as x and y approach zero?Proving inexistence of limitMultivariable limit of rational functionCalculate double limit of $x^2sinfrac1xy$Difficulty proving multivariate limit involving sin^2(x) does not existProof that the limit exists using polar coordinatesProving Multivariate Limit with Squeeze TheoremTwo-variable limit of $lim_(x,y)to(0,0)fracsin(x^4+y^4)x^2+y^2$Calculate if this Multivariable Limit existhow to prove limit exists or not?

1

1

$begingroup$

I'm trying to prove that the limit:

$lim_(x,y)to (0,0)|x|^y=1$

I started with
$lim_(x,y)to (0,0)^y=lim_(x,y)to (0,0)e^=e^lim_(x,y)to (0,0)$

And then tried substituting $x=rsintheta$ and $y=rcostheta$ to get:

$e^lim_rto 0rcostheta$

However, as I'm trying to use the squeeze theorem, I can't figure out a lower bound that would work on this. Any suggestions? Does the limit even exist?

limits multivariable-calculus

share|cite|improve this question

edited Mar 17 at 13:17

YuiTo Cheng

2,1362837

asked Mar 17 at 12:35

user653788user653788

102

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  See my answer to Multivariable limits.
  $endgroup$
  – Dave L. Renfro
  Mar 17 at 12:43
  
  

  
  
  
  

add a comment | 

1

1

$begingroup$

I'm trying to prove that the limit:

$lim_(x,y)to (0,0)|x|^y=1$

I started with
$lim_(x,y)to (0,0)^y=lim_(x,y)to (0,0)e^=e^lim_(x,y)to (0,0)$

And then tried substituting $x=rsintheta$ and $y=rcostheta$ to get:

$e^lim_rto 0rcostheta$

However, as I'm trying to use the squeeze theorem, I can't figure out a lower bound that would work on this. Any suggestions? Does the limit even exist?

limits multivariable-calculus

share|cite|improve this question

edited Mar 17 at 13:17

YuiTo Cheng

2,1362837

asked Mar 17 at 12:35

user653788user653788

102

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  See my answer to Multivariable limits.
  $endgroup$
  – Dave L. Renfro
  Mar 17 at 12:43
  
  

  
  
  
  

add a comment | 

$begingroup$

I'm trying to prove that the limit:

$lim_(x,y)to (0,0)|x|^y=1$

I started with
$lim_(x,y)to (0,0)^y=lim_(x,y)to (0,0)e^=e^lim_(x,y)to (0,0)$

And then tried substituting $x=rsintheta$ and $y=rcostheta$ to get:

$e^lim_rto 0rcostheta$

However, as I'm trying to use the squeeze theorem, I can't figure out a lower bound that would work on this. Any suggestions? Does the limit even exist?

limits multivariable-calculus

share|cite|improve this question

edited Mar 17 at 13:17

YuiTo Cheng

2,1362837

asked Mar 17 at 12:35

user653788user653788

102

$endgroup$

share|cite|improve this question

edited Mar 17 at 13:17

YuiTo Cheng

2,1362837

asked Mar 17 at 12:35

user653788user653788

102

share|cite|improve this question

edited Mar 17 at 13:17

YuiTo Cheng

2,1362837

asked Mar 17 at 12:35

user653788user653788

102

edited Mar 17 at 13:17

YuiTo Cheng

2,1362837

edited Mar 17 at 13:17

YuiTo Cheng

2,1362837

asked Mar 17 at 12:35

user653788user653788

102

asked Mar 17 at 12:35

user653788user653788

102

2 Answers
2

active

oldest

votes

2

$begingroup$

This limit doesn't exist. Consider two different paths $y=0$ and $x=0$. One leads to $0$-limit and the other to $1$-limit. The figure below shows why:

share|cite|improve this answer

answered Mar 17 at 13:10

Mostafa AyazMostafa Ayaz

17.8k31039

$endgroup$

add a comment | 

2

$begingroup$

The limit does not exist: When approaching $(0,0)$ along the $x$ axis we have $|x|^y=|x|^0=1$, while when approaching along the $y$ axis we have $|x|^y=0^y=0$.

share|cite|improve this answer

edited Mar 17 at 13:06

answered Mar 17 at 12:56

Barry CipraBarry Cipra

60.5k655128

$endgroup$

add a comment | 

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2

$begingroup$

This limit doesn't exist. Consider two different paths $y=0$ and $x=0$. One leads to $0$-limit and the other to $1$-limit. The figure below shows why:

share|cite|improve this answer

answered Mar 17 at 13:10

Mostafa AyazMostafa Ayaz

17.8k31039

$endgroup$

add a comment | 

2

$begingroup$

This limit doesn't exist. Consider two different paths $y=0$ and $x=0$. One leads to $0$-limit and the other to $1$-limit. The figure below shows why:

share|cite|improve this answer

answered Mar 17 at 13:10

Mostafa AyazMostafa Ayaz

17.8k31039

$endgroup$

add a comment | 

$begingroup$

This limit doesn't exist. Consider two different paths $y=0$ and $x=0$. One leads to $0$-limit and the other to $1$-limit. The figure below shows why:

share|cite|improve this answer

answered Mar 17 at 13:10

Mostafa AyazMostafa Ayaz

17.8k31039

$endgroup$

share|cite|improve this answer

answered Mar 17 at 13:10

Mostafa AyazMostafa Ayaz

17.8k31039

answered Mar 17 at 13:10

Mostafa AyazMostafa Ayaz

17.8k31039

answered Mar 17 at 13:10

Mostafa AyazMostafa Ayaz

17.8k31039

2

$begingroup$

The limit does not exist: When approaching $(0,0)$ along the $x$ axis we have $|x|^y=|x|^0=1$, while when approaching along the $y$ axis we have $|x|^y=0^y=0$.

share|cite|improve this answer

edited Mar 17 at 13:06

answered Mar 17 at 12:56

Barry CipraBarry Cipra

60.5k655128

$endgroup$

add a comment | 

2

$begingroup$

The limit does not exist: When approaching $(0,0)$ along the $x$ axis we have $|x|^y=|x|^0=1$, while when approaching along the $y$ axis we have $|x|^y=0^y=0$.

share|cite|improve this answer

edited Mar 17 at 13:06

answered Mar 17 at 12:56

Barry CipraBarry Cipra

60.5k655128

$endgroup$

add a comment | 

$begingroup$

The limit does not exist: When approaching $(0,0)$ along the $x$ axis we have $|x|^y=|x|^0=1$, while when approaching along the $y$ axis we have $|x|^y=0^y=0$.

share|cite|improve this answer

edited Mar 17 at 13:06

answered Mar 17 at 12:56

Barry CipraBarry Cipra

60.5k655128

$endgroup$

share|cite|improve this answer

edited Mar 17 at 13:06

answered Mar 17 at 12:56

Barry CipraBarry Cipra

60.5k655128

answered Mar 17 at 12:56

Barry CipraBarry Cipra

60.5k655128

answered Mar 17 at 12:56

Barry CipraBarry Cipra

60.5k655128