# Limit in multivariable calculus

1. May 1, 2013

### Gauss M.D.

1. The problem statement, all variables and given/known data

Examine lim (x,y) -> (0,0) of:

$\sqrt{x^{2}+1} - \sqrt{y^{2}-1}$

$\frac{\sqrt{x^{2}+1} - \sqrt{y^{2}-1}}{x^{2}+y^{2}}$

3. The attempt at a solution

Tried variable sub:

$\sqrt{x^{2}+1} = a, \sqrt{y^{2}-1} = b$

$\frac{a - b}{a^{2}-b^{2}}$

(a -> 1, b -> 1 as x,y -> 0)

Still nasty

Tried polar coordinates:

$\sqrt{1 + r^{2}cosθ} - \sqrt{1 - r^{2}sinθ}/r^{2}$

But I can't find a way for this limit to exist (which it is supposed to do according to the answers).

2. May 1, 2013

### Gauss M.D.

I think I got it. Conjugate rule.....

3. May 1, 2013

### SammyS

Staff Emeritus
Are you sure there's not some typo here?

$\displaystyle \sqrt{y^2-1\,}\$ is only defined for |y| ≥ 1 .

4. May 1, 2013

### Gauss M.D.

Yeah, meant to write 1-y^2. But I figured it out anyway :)

5. May 1, 2013

### Gauss M.D.

Although I still have something I want to make clear: Say I wanted to show that a particular limit does NOT exist, for example:

$xy^{2}/(x^{2}+y^{4})$

Is the following valid?

Let y = x:

$x^{3}/(x^{2}+x^{4}) = x/1+x^{2}$

Which is 0 as x,y -> 0

Now let y = $\sqrt{x}$

$x^{2}/(2x^{2})$

Since I have found a path to the origin with an undefined limit, does that mean I have proven that the original limit does not exist?

Sorry, I meant, is it sufficient to find two curves with DIFFERENT limits?

I.e. let y = x and you have the limit = 0, and y = sqrt(x) and the limit = 1/2

Last edited by a moderator: May 4, 2013
6. May 1, 2013

### SammyS

Staff Emeritus
My first comment: Be careful with parentheses. You have unbalanced parentheses in one of your expressions. One of your other expression has a more serious problem. It needs an additional set of parentheses to make it true.
What this says literally is that $\ \displaystyle x^{3}/(x^{2}+x^{4}) = \frac{x}{1}+x^{2}\ .$

What you meant to say (I hope) is $\ \displaystyle x^{3}/(x^{2}+x^{4}) = x/\left(1+x^{2}\right)\,,\$ which is equivalent to $\ \displaystyle x^{3}/(x^{2}+x^{4}) = \frac{x}{1+x^{2}}\ .$
This should have been $\ x^{2}/(2x^{2})\ .$

If the limit along that second path was indeed undefined, then yes, that would show that the original limit does not exist.

$\displaystyle \lim_{x\to\,0} \left(\frac{x^{2}}{2x^{2}}\right)\$ does exist. What is it?

(It's different than the limit along the path y = x. That does show that the original limit does not exist.)

7. May 1, 2013

### Gauss M.D.

Yeah that was actually exactly what I meant but got confused in my latex-noobness :S