Find $\lim\limits_{x\to 0}\frac{e^{x^2}-\cos x^{\sqrt{2}}}{x^2}$ 
Find $\lim\limits_{x\to 0}\frac{e^{x^2}-\cos x^{\sqrt{2}}}{x^2}$

Using Taylor series:
$$\lim\limits_{x\to 0}\frac{e^{x^2}-\cos x^{\sqrt{2}}}{x^2}=\lim\limits_{x\to 0}\frac{(1+x^2+O(x^{2n}))-(1-\frac{x^{2\sqrt{2}}}{2}+O(x^{\sqrt{2}(2n+1)})}{x^2}=\lim\limits_{x\to 0}\frac{\frac{2x^2+x^{2\sqrt{2}}}{2}}{x^2}=\lim\limits_{x\to 0}\frac{2x^2+x^{2\sqrt{2}}}{2x^2}$$
How to evaluate this limit?

Hint. You may just write, as $x \to 0$,
$$
\frac{e^{x^2}-\cos (x^{\sqrt{2}})}{x^2}=\frac{x^2(1+\frac12x^{2\sqrt{2}-2})+O(x^4)}{x^2}=1+\frac12x^{2\sqrt{2}-2}+O(x^2) \longrightarrow 1.
$$
In fact, $x^{2\sqrt{2}}$ is negligible comparing to $x^2$.
