How is the limit of this function calculated

Question

The function $f$ is defined as:

$$f(a) = \frac{\sigma}{\sqrt{2a}} \sqrt{1-e^{-2as}} Q_p$$

I want to calculate the limit $ lim_{a\to0} f(a)$. The function is in indeterminate $\frac00$ form. The answer in the text is given as: $$\lim_{a\to0} f(a)= \sigma\sqrt{s}Q_p$$

However, when I calculate it using L'Hospital, I get: $$\lim_{a\to0} f(a) = \sigma\frac{s}{2 \sqrt{1-e^{-2as}}} Q_p$$ Where is my mistake? How do I get the correct answer?

Answer 1

Hint:

The easiest thing to do is to take all constants aside and just evaluate the limit inside the square root, since $\sqrt x$ is continuous. $$\frac{\sigma Q_p}{\sqrt 2} \sqrt{ \lim_{a\to 0} \frac{1-e^{-2as}}{a}}$$

Use L.H. to finish.

Answer 2

What is $\lim_{x\rightarrow 0} \frac{\exp(x)-1}{x}$ ? Once you know this, you can easily conclude.