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February 24th, 2017, 03:08 PM   #1
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Question Limits of Functions of Two Variables

In Calculus III, we've covered limits of functions of two variables, but I'm still somewhat confused by the topic. The limit has been defined to only exist if the limit approaches the same value from "all possible paths" along the surface. However, we never definitively showed how to test "all possible paths." We could show that a limit might not exist at a particular point by evaluating the limit along different paths and observing that the values of those limits are different, but showing that a limit doesn't exist at a point is much easier than showing that it does.

So for any given function of two variables, what is the process to show that a limit exists for sure at a point on the surface (or a point where the function is undefined)?
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February 24th, 2017, 05:45 PM   #2
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If the function is continuous at the point, you can just plug in the values.

Otherwise you could convert to polars.

If you want a more rigorous proof you can use the precise definition:

If for all $\displaystyle \begin{align*} \epsilon > 0 \end{align*}$ there exists a $\displaystyle \begin{align*} \delta > 0 \end{align*}$ such that $\displaystyle \begin{align*} \sqrt{ \left( x - a \right) ^2 + \left( y - b \right) ^2 } < \delta \implies \left| f\left( x, y \right) - L \right| < \epsilon \end{align*}$ then $\displaystyle \begin{align*} \lim_{\left( x, y \right) \to \left( a , b \right) } f\left( x,y \right) = L \end{align*}$.
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February 24th, 2017, 05:51 PM   #3
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In other words, there is no distinct method that can be applied to any function to determine whether or not a limit exists.
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February 24th, 2017, 06:25 PM   #4
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Okay, so let's take this, for example:

So if we're to show that this piecewise function is continuous, we have to show that the limit as (a,b) approaches (0,0) of the top portion of the piecewise function is zero from all directions. I'm still not sure though; how exactly would I do that?
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February 24th, 2017, 07:45 PM   #5
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Quote:
Originally Posted by John Travolski View Post
Okay, so let's take this, for example:

So if we're to show that this piecewise function is continuous, we have to show that the limit as (a,b) approaches (0,0) of the top portion of the piecewise function is zero from all directions. I'm still not sure though; how exactly would I do that?
convert to polars

$\begin{cases}\dfrac{r^2 \sin(2\theta)}{2 r^2} &r \neq 0 \\0 &r=0\end{cases} = $

$\begin{cases}\dfrac{ \sin(2\theta)}{2} &r \neq 0 \\0 &r=0 \end{cases}$

This will have a continuum of values as $0 \leq \theta < 2\pi$ and thus there will be no limit at $(0,0)$
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February 25th, 2017, 08:31 AM   #6
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Okay, I think that I understand now. Thank you.
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February 25th, 2017, 01:11 PM   #7
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More easily, approach the origin along the line $y=kx$ and we clearly approach $\frac{k}{1+k^2}$ which is equal to zero only when $k=0$.
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Last edited by v8archie; February 25th, 2017 at 01:22 PM.
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