Question

Estimate the value of the limit: lim[x->0] (e^x-x-1)/x^2

How can I simplify this?

Answer 1

I suppose you can use L'Hopital Rule here.

Answer 2

We can also use Taylor series.
\[\begin{align*}\frac{e^x-x-1}{x^2}&=\frac{1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots-x-1}{x^2}\\\\&=\frac{\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots}{x^2}\\\\
&=\frac{1}{2!}+\frac{x}{3!}+\cdots\end{align*}\]
What happens as \(x\to0\)?