A limit with the exponential

#limits#taylor

Statement

Compute $\displaystyle\lim_{x\to0}\frac{e^x-1-x}{x^2}$ .

Hint

Use the Taylor expansion of $e^x$ at $0$ up to order $2$ .

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As $x\to0$ , $e^x=1+x+\dfrac{x^2}{2}+o(x^2)$ .

Therefore $\dfrac{e^x-1-x}{x^2}=\dfrac12+o(1)$ , and the limit is $\dfrac12$ .