# Multi-variable Limit

Question: Does the following limit exist, if so what does it equal? $$\lim_{(x, y) \to (0,0)} \frac{x^2 y^5}{2x^4 +3y^{10}}$$

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Solution 1: The limit DOES NOT exist.

Let $x=y^{5/2}$ $$\lim_{y \to 0} \frac{(y^{5/2})^2 y^5}{2(y^{5/2})^4 +3y^{10}} = \lim_{y \to 0} \frac{y^{10}}{2y^{10} +3y^{10}} = \lim_{y \to 0} \frac{1}{2 +3} = \frac{1}{5}$$

Let $x=0$ $$\lim_{y \to 0} \frac{(0)^2 y^5}{2(0)^4 +3y^{10}} = \lim_{y \to 0} \frac{0}{0 +3y^{10}} = \lim_{y \to 0} 0 = 0$$

Thus the limit does not exist

Solution 2: The limit DOES exist.

Change to polar coordinates and find limit as $r \to 0$ $$\lim_{r \to 0} \frac{(r^2cos^2{\theta}) (r^5sin^5\theta)}{2(r^4cos^4{\theta}) +3(r^{10}cos^{10}{\theta})} =\lim_{r \to 0}\frac{r^4}{r^4} \times\frac{r^3cos^2{\theta} sin^5\theta}{2cos^4{\theta} +3r^{6}cos^{10}{\theta}} =\frac{0}{2cos^4{\theta} +0}= 0$$

Thus the limit does exist and is $0$

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Problem Can someone please tell me which solution is correct (if any) and why the other is wrong?

The first solution is correct, the limit does not exists. The second one is wrong because the "limit" $$\frac{0}{0 + 2 \cos^4\theta}$$ is not well-defined if $\theta = \frac\pi 2$.