EDIT: Problem solved; taking $\displaystyle \delta =\frac{12}{\epsilon}$ is (of course) just a mistake. However, I guess this thread still serves as a good exercise in proving multivariable limits by using polar coordinates. By the way, the original function in x and y is given by $\displaystyle f(x,y)=\frac{3(x^4+y^6-x^3y^4)}{x^4+y^6}$.

Consider the following solution to an exercise in which one is to give a proof using polar coordinates to prove the limit of a function at $\displaystyle (x,y)=(0,0)$

Given

$\displaystyle

f(rcos(\theta),rsin(\theta))={\frac {3\,{r}^{4} \left( \cos \left( \theta \right) \right) ^{4}+3\,

{r}^{6} \left( \sin \left( \theta \right) \right) ^{6}-3\,{r}^{7}

\left( \cos \left( \theta \right) \right) ^{3} \left( \sin \left(

\theta \right) \right) ^{4}}{{r}^{4} \left( \cos \left( \theta

\right) \right) ^{4}+{r}^{6} \left( \sin \left( \theta \right)

\right) ^{6}}}

$

we want to show that the function $\displaystyle f$ has limit$\displaystyle L=3$ as $\displaystyle r$ approaches zero.

Assuming $\displaystyle r\leq 1$, we can establish the following,

$\displaystyle

\left| f \left( r\cos \left( \theta \right) ,r\sin \left( \theta

\right) \right) -3 \right|\leq 3\,{\frac {{r}^{7}}{{r}^{4} \left( \cos \left( \theta \right)

\right) ^{4}+{r}^{6} \left( \sin \left( \theta \right) \right) ^{6}}

}

$

$\displaystyle

\Rightarrow \left| f \left( r\cos \left( \theta \right) ,r\sin \left( \theta

\right) \right) -3 \right|\leq 3\,{\frac {{r}^{7}}{{r}^{6} \left( \cos \left( \theta \right) \right) ^{6}+{r}^{6} \left( \sin \left( \theta \right) \right) ^{6}}}

$

Further, noting that$\displaystyle \left( \cos \left( \theta \right) \right) ^{6}+ \left( \sin \left(

\theta \right) \right) ^{6}\geq \frac {1}{4}

$, we have finally

$\displaystyle

\left| f \left( r\cos \left( \theta \right) ,r\sin \left( \theta

\right) \right) -3 \right|\leq 12r

$

Thus we may define$\displaystyle \delta =\frac{12}{\epsilon}$so that,$\displaystyle

0 < r < \delta =\frac{12}{\epsilon} \Rightarrow \left| f(rcos(\theta),rsin(\theta)) - 3\right| < \epsilon

$

Now, here's my problem; I'm perfectly fine with the reasoning here up until the conclusion where $\displaystyle \delta$ is taken to be $\displaystyle \frac{12}{\epsilon}$. To me it is fairly obvious that, instead, letting $\displaystyle \delta =\frac{\epsilon}{12}$ is a valid choice in establishing this proof, and that's what I've been teaching when instructing a few of the students who take a basic course in multivariable calculus at my university up until I was emailed this solution from a professor who teach the subject.

I don't see why taking $\displaystyle \delta =\frac{12}{\epsilon}$ could be correct, since with $\displaystyle r \leq 1$ we'd be forced to take $\displaystyle \epsilon > 12$, but according to the definition we should be able to choose an arbitrary $\displaystyle \epsilon > 0$.

So, am I missing something here? Is there any reason why the first choice of $\displaystyle \epsilon$ should be valid or is it simply a typo?

Would be nice if someone could comment on this so that I can know if I'm really running around lying to the students around here or not.