<s>[INST] 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?
 [/INST] Starting from where you left off, we divide the numerator and denominator by $x^2$ we obtain $\frac{2+x^{2\sqrt{2}-2}}{2}$. Now $\lim_{x\to 0}{\frac{2+x^{2\sqrt{2}-2}}{2}}=\frac{2+0}{2}=1$
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