<s>[INST] Find $\lim_{(x,y)\rightarrow(0,0)} \cos(\frac{x^2-y^2}{\sqrt{x^2+y^2}})$ $\lim_{(x,y)\rightarrow(0,0)} \cos(\frac{x^2-y^2}{\sqrt{x^2+y^2}})$
I tried replacing x and y with several values and kept getting 1 so I tried:
$$0 \le |\cos(\frac{x^2-y^2}{\sqrt{x^2+y^2}})| \le |\cos(|(\frac{x^2-y^2}{\sqrt{x^2+y^2}}|)|\le |\cos(\frac{|x|^2+|y|^2}{\sqrt{x^2+y^2}})|\le |\cos(\frac{2\sqrt{x^2+y^2}^2}{\sqrt{x^2+y^2}})| \le |\cos(2\sqrt{x^2+y^2})|$$
Is this correct? I couldn't get Wolfram to compute this properly for some reason.
 [/INST] HINT
Consider the polar coordinates $x = r\cos(\theta)$ and $y = r\sin(\theta)$. Then we get
\begin{align*}
\lim_{(x,y)\rightarrow(0,0)}\cos\left(\frac{x^{2}-y^{2}}{\sqrt{x^{2}+y^{2}}}\right) = \lim_{r\rightarrow 0}\cos(r\cos(2\theta)) = \ldots
\end{align*}
Can you take it from here?
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