I want to evaluate the following limit:
For $x\rightarrow0$, the denominator is asymptotic to
Here's how I simplify the numerator:
Finally we have
My textbook does it another way and the solution is 2. I wonder what I'm doing wrong. Perhaps it's the fact that I have replaced $e^x$ with $1$ but that seemed to make sense to me because $e^x\rightarrow1$ for $x\rightarrow0$. Any hints?