# 2-variable limit

• Sep 6th 2011, 09:25 AM
marqushogas
2-variable limit

[x^5 + y^2]/[x^4 + |y|] -> 0 as (x,y) -> (0,0)
• Sep 6th 2011, 09:30 AM
Prove It
Re: 2-variable limit
Quote:

Originally Posted by marqushogas

[x^5 + y^2]/[x^4 + |y|] -> 0 as (x,y) -> (0,0)

Maybe converting to polars might help?

$\displaystyle \displaystyle x = r\cos{\theta}$ and $\displaystyle \displaystyle y = r\sin{\theta}$, so

\displaystyle \displaystyle \begin{align*} \lim_{(x, y) \to (0, 0)}\frac{x^5 + y^2}{x^4 + |y|} &= \lim_{r \to 0}\frac{r^5\cos^5{\theta} + r^2\sin^2{\theta}}{r^4\cos^4{\theta} + |r||\sin{\theta}|} \\ &= \lim_{r \to 0}\frac{r^2\left(r^3\cos^5{\theta} + \sin^2{\theta}\right)}{|r|\left(|r|r^2 \cos^4{\theta} + | \sin{\theta} | \right)} \\ &= \lim_{r \to 0}\frac{|r|\left(r^3\cos^5{\theta} + \sin^2{\theta} \right)}{|r|r^2\cos^5{\theta} + | \sin{\theta} |} \\ &= \frac{0\left( 0 + \sin^2{\theta} \right)}{0 + | \sin{\theta} |} \\ &= \frac{0}{|\sin{\theta}|} \\ &= 0 \end{align*}
• Sep 6th 2011, 11:58 AM
FernandoRevilla
Re: 2-variable limit
Quote:

Originally Posted by Prove It
$\displaystyle = \frac{0}{|\sin{\theta}|}= 0$

$\displaystyle \frac{0}{|\sin{\theta}|}= 0 \Leftrightarrow \sin \theta\neq 0$ and $\displaystyle \sin \theta =0$ iff $\displaystyle (x,y)$ belongs to the set $\displaystyle S=\{(x,0):x\neq 0\}$ . Now, $\displaystyle f(x,y)=x$ on $\displaystyle S$ and the limit of $\displaystyle f(x,y)$ along $\displaystyle S$ for $\displaystyle (x,y)\to (0,0)$ is also $\displaystyle 0$ .