<s>[INST] How to find the value of $\lim_\limits{x \to 0}\frac{e^{x\cos{x^2}}-e^{x}}{x^5}$ How to find the exact value of the following limit?
$\lim_\limits{x \to 0}\frac{e^{x\cos{x^2}}-e^{x}}{x^5}$
 [/INST] $$\cos x^2=1-\frac{x^4}{2}+o(x^4)$$
so
$$e^{x\cos x^2} = e^{x-\frac{x^5}{2}+o(x^5)} = e^xe^{-\frac{x^5}{2}+o(x^5)}$$
therefore 
$$\frac{e^{x\cos x^2}-e^x}{x^5} = e^x\frac{e^{-\frac{x^5}{2}+o(x^5)}-1}{x^5} \sim e^x\frac{-\frac{x^5}{2}}{x^5} \xrightarrow[x\to0]{}-\frac{1}{2}$$
using $e^u-1\sim u$ when $u\to0$.
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