Does the Multivariable Limit of xy^8/(x^3+y^12) Exist?

In summary, my friend's lecturer is adamant that the multivariable limit does not exist, but when we converted to polars and let r approach 0, we still approached the same value as $r \to 0$.
  • #1
Prove It
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Hi everyone. A friend of mine asked for help evaluating this multivariable limit.

$\displaystyle \begin{align*} \lim_{(x,y) \to (0,0)} \frac{x\,y^8}{x^3 + y^{12}} \end{align*}$

We got the answer of 0 by converting to polars.

$\displaystyle \begin{align*} \lim_{(x,y) \to (0,0)} \frac{x\,y^8}{x^3 + y^{12}} &= \lim_{r \to 0} \frac{r\cos{( \theta )} \, \left[ r\sin{ (\theta ) } \right] ^8}{ \left[ r\cos{ (\theta ) } \right] ^3 + \left[ r\sin{ ( \theta ) } \right] ^{12} } \\ &= \lim_{r \to 0} \frac{r^9 \cos{ ( \theta ) } \sin^8{ ( \theta ) } }{r^3 \cos^3{( \theta ) } + r^{12} \sin^{12}{ ( \theta ) } } \\ &= \lim_{r \to 0} \frac{r^6 \cos{( \theta ) } \sin^8{ ( \theta ) } }{ \cos^3{( \theta )} + r^9 \sin^{12}{(\theta ) } } \\ &= \frac{0}{\cos^3{(\theta ) } } \end{align*}$

Now for all $\displaystyle \begin{align*} \theta \neq \frac{ \left( 2n + 1 \right) \, \pi}{2}, n \in \mathbf{Z} \end{align*}$, we have $\displaystyle \begin{align*} \frac{0}{\cos^3{( \theta ) }} = 0 \end{align*}$ and since $\displaystyle \begin{align*} \theta = \frac{ \left( 2n + 1 \right) \, \pi}{2} \end{align*}$ are all REMOVABLE discontinuities (i.e. holes), that means $\displaystyle \begin{align*} \lim_{\theta \to \frac{ \left( 2n + 1 \right) \, \pi}{2}} \frac{0}{\cos^3{( \theta ) }} = 0 \end{align*}$ as well, thereby still approaching the same value from those paths too...

Therefore surely $\displaystyle \begin{align*} \lim_{(x,y) \to (0,0) } \frac{x\,y^8}{x^3 + y^{12}} = 0 \end{align*}$.

Wolfram appears to agree with us too.My friend's online assessment however has the answer of DNE. When asked about it and showing our work and Wolfram's output, my friend's lecturer is still adamant that the limit does not exist, because the value we get when converting to polars and letting r approach 0 isn't always defined i.e. where $\displaystyle \begin{align*} \theta = \frac{ \left( 2n + 1 \right)\, \pi}{2} \end{align*}$. But wouldn't that just mean that there's more paths that need to be tested, and as we have shown, we still approach the same value as $\displaystyle \begin{align*} r \to 0 \end{align*}$ from all paths with $\displaystyle \begin{align*} \theta \neq \frac{ \left( 2n + 1 \right) \, \pi}{2} \end{align*}$ as we do when $\displaystyle \begin{align*} \theta = \frac{ \left( 2n + 1 \right) \, \pi}{2} \end{align*}$.Would anyone here be able to go over our work and verify who is correct in this case please?
 
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  • #2
Your friend's lecturer is correct. Consider the path $x=y^4$. Then

$$\frac{xy^8}{x^3+y^{12}} = \frac{y^4 y^8}{(y^4)^3 + y^{12}} = \frac{y^{12}}{2y^{12}} = \frac{1}{2},$$

and the limit is $1/2$.

Best wishes. :)
 
  • #3
Fantini said:
Your friend's lecturer is correct. Consider the path $x=y^4$. Then

$$\frac{xy^8}{x^3+y^{12}} = \frac{y^4 y^8}{(y^4)^3 + y^{12}} = \frac{y^{12}}{2y^{12}} = \frac{1}{2},$$

and the limit is $1/2$.

Best wishes. :)

A beautiful counterexample, how did you come up with it? :)
 
  • #4
Hi Prove It,

Since Fantini has already displayed a counterexample, I'm giving a small remark that I hope will help. Looking at your work, I see that after changing to polar coordinates, you assumed $\theta$ is constant and found the limit to be zero. However, finding a bivariate limit by changing to polar coordinates does not always work, since it's possible to have the limit exist along every ray through the origin, but fail to exist in the general sense.

Here's a rigorous statement regarding the polar method. Let $f : \Bbb R^2 \to \Bbb R$. Suppose there is an $L$ such that to every $\epsilon > 0$, there corresponds a $\delta > 0$ such that for all $r$ and $\theta$, $|r| < \delta$ implies $|f(r,\theta) - L| < \epsilon$. Then

\(\displaystyle \lim_{(x,y)\to (0,0)} f(x,y) = \lim_{r\to 0} f(r,\theta) = L.\)
 
  • #5
This is a variation of a known limit that seemingly exists for all lines, such as when you tested polar coordinates, but when tested along other curves (like parabolas or the sort) does not exist. :) The common form is

$$\frac{xy^2}{x^2+y^4}.$$

Thank you for the compliment. Best wishes. :)
 

FAQ: Does the Multivariable Limit of xy^8/(x^3+y^12) Exist?

What is a multivariable limit?

A multivariable limit is a mathematical concept that deals with the behavior of a function as it approaches a specific point in a multidimensional space. It is used to determine the value that a function approaches as the independent variables approach a specific point.

Why is the concept of multivariable limit important?

The concept of multivariable limit is important because it allows us to analyze the behavior of functions in multiple dimensions, which is crucial in many fields such as physics, economics, and engineering. It also helps us understand the continuity and differentiability of functions in multidimensional spaces.

How is a multivariable limit different from a single variable limit?

A single variable limit deals with the behavior of a function as it approaches a specific point on a one-dimensional number line. In contrast, a multivariable limit deals with the behavior of a function as it approaches a specific point in a multidimensional space, taking into account multiple independent variables.

What are the common methods used to evaluate multivariable limits?

The common methods used to evaluate multivariable limits include substitution, converting to polar or spherical coordinates, and using the squeeze theorem. In some cases, it may also be necessary to use L'Hôpital's rule or to break the limit into one-dimensional limits.

What are the common misconceptions about multivariable limits?

One common misconception is that multivariable limits are always equal to the value of the function at the point in question. However, this is only true in certain cases, and in general, the limit may not exist or may be different from the value of the function. Another misconception is that multivariable limits are only relevant in mathematical contexts, when in fact they have practical applications in many fields of science and engineering.

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