# Thread: limit of multi variable functions

1. ## limit of multi variable functions

So we were learning about 2 dimensional domain functions today, and my prof put up a limit. he did some work and then did something I was a bit stumped on for a while: $\displaystyle lim_{y\to\0} \frac{0}{y^2} \Rightarrow 0$. Isn't this indeterminate? Would it be 0 because y^2 is not ACTUALLY 0, but simple a number really close to 0 and then numerator is fixed at 0, so 0 divided by any number close to 0 is still 0. Am I right?

P.S. How do you get $\displaystyle y \rightarrow 0$ to go under the word "limit" rather than next to it?

2. ## Re: limit of multi variable functions

use \displaystyle

\displaystyle{\lim_{y \to 0}} ~\dfrac {0}{y^2}

$\displaystyle{\lim_{y \to 0}} ~\dfrac {0}{y^2}$

Maybe one of the harder core math gurus here will chime in on the math of this.

Note that with 2 applications of L'Hopital's rule you end up with

$\displaystyle{\lim_{y \to 0}}~\dfrac 0 2 = 0$ in a form that's well defined.

It's easy enough to see though that you can get $\dfrac 0 {y^2}$ arbitrarily close to zero (it's always zero) without y ever having to actually be zero thus making the quotient undefined.

3. ## Re: limit of multi variable functions

Originally Posted by toesockshoe
So we were learning about 2 dimensional domain functions today, and my prof put up a limit. he did some work and then did something I was a bit stumped on for a while: $\displaystyle \lim_{y\to\0} \frac{0}{y^2} \Rightarrow 0$. Isn't this indeterminate? Would it be 0 because y^2 is not ACTUALLY 0, but simple a number really close to 0 and then numerator is fixed at 0, so 0 divided by any number close to 0 is still 0. Am I right?

P.S. How do you get $\displaystyle y \rightarrow 0$ to go under the word "limit" rather than next to it?
Look at the quote above the $y\to 0$ is correctly placed using \lim_{y\to\0} \frac{0}{y^2} , I added a \.

You have ignored an important point in any $\displaystyle \lim_{y\to a} f(y)$ it must be the case that $y\ne a$ always.

If $y\ne 0$ then $\dfrac{0}{y^2}=0$ therefore the limit is $0$.

4. ## Re: limit of multi variable functions

there appear to be differences in which LaTex interpreter you choose to use.

$\displaystyle \lim_{y \to 0} \dfrac 0 {y^2}$

$\lim_{y \to 0} \dfrac 0 {y^2}$

$\displaystyle \lim_{y \to \0} \dfrac 0 {y^2}$

$\lim_{y \to \0} \dfrac 0 {y^2}$

$\displaystyle{\lim_{y \to 0}} \dfrac 0 {y^2}$

5. ## Re: limit of multi variable functions

Originally Posted by toesockshoe
Would it be 0 because y^2 is not ACTUALLY 0, but simple a number really close to 0 and then numerator is fixed at 0, so 0 divided by any number close to 0 is still 0. Am I right?
Yes.