Answer
The solution is
$$\lim_{x\to0}\frac{e^x-x-1}{5x^2}=\frac{1}{10}.$$
Work Step by Step
We will calculate this limit using L'Hopital's rule. "LR" will stand for "Apply L'Hopital's rule".
$$\lim_{x\to0}\frac{e^x-x-1}{5x^2}=\left[\frac{0}{0}\right][\text{LR}]=\lim_{x\to0}\frac{(e^x-x-1)'}{(5x^2)'}=\lim_{x\to0}\frac{e^x-1}{10x}=\left[\frac{e^0-1}{10\cdot1}\right]=\left[\frac{0}{0}\right][\text{LR}]=\lim_{x\to0}\frac{(e^x-1)'}{(10x)'}=\lim_{x\to0}\frac{e^x}{10}=\frac{e^0}{10}=\frac{1}{10}.$$
