<s>[INST] Determine whether a multi-variable limit exists $\lim_{(x,y)\to(0,0)}\frac{\cos x-1-\frac{x^2}2}{x^4+y^4}$ I need to determine whether the next limit exists: 
$$\lim_{(x,y)\to(0,0)}f(x,y)=\lim_{(x,y)\to(0,0)}\frac{\cos x-1-\frac{x^2}2}{x^4+y^4}$$
Looking at the numerator $(-1-\frac{x^2}2)$ it immediately reminds me of maclaurin series of $\cos$, Where:
$$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}+...=1-\frac{x^2}{2!}+\frac{x^4}{4!}+O(x^6)$$
So we can simplify the expression:
$\begin{align} \lim_{(x,y)\to(0,0)}\frac{\cos x-1-\frac{x^2}2}{x^4+y^4}&=\lim_{(x,y)\to(0,0)}\frac{1-\frac{x^2}{2!}+\frac{x^4}{4!}+O(x^6)-1-\frac{x^2}2}{x^4+y^4}\\
&=\lim_{(x,y)\to(0,0)}\frac{-x^2+\frac{x^4}{4!}+O(x^6)}{x^4+y^4} \end{align}$
This limit does not exist.
So, after solving it the 'hard' way, I was thinking:
If I place $y=mx$ and then try to solve the limit I get:
$$\lim_{(x,y)\to(0,0)}\frac{\cos x-1-\frac{x^2}2}{x^4(1+m^4)}=\infty$$
Is that enough to prove that the limit does not exists? I mean, If I find at least one limit that does not 'converge' is it enough to say that a multi-variable function limit doesn't exist?
 [/INST] HINT
Find the limit as $x \to 0$ when $y=x$ and show that it is not equal the limit as $x\to 0$ when $y=0$.
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