Computing the limit of an expression I have the following question. Determine the limit of the following expression.
$\lim_{x \rightarrow 0} \frac{cos(x \sqrt{2}) - \frac{1}{1+x^2}}{x^4}$.
My attempt to this question is the following. Let the function $f(x) = \frac{cos(x \sqrt{2}) - \frac{1}{1+x^2}}{x^4} = \frac{\cos(x \sqrt{2})}{x^4} - \frac{1}{(1+x^2) x^4}.$ Then $\lim_{x \rightarrow 0} f(x) = \lim_{x \rightarrow 0} \frac{\cos(x \sqrt{2})}{x^4} - \lim_{x \rightarrow 0} \frac{1}{(1+x^2) x^4} = 0 - Undefined$. So the limit does not exists.
But my professor's answer is $-\frac{5}{6}$. He used Taylor expansion. What am I doing wrong??

we get by simplifying your term
$\frac{(1+x^2)\cos(x\sqrt{2})-1)}{x^4(1+x^2)}$ and the taylor expansion for $x=0$ is $-\frac{5}{6}+O(x^2)$
