TITLE: Finding the limit of a difficult exponential function
QUESTION [12 upvotes]: I am stumped on this question, any help or tips would be appreciated!
Find the limit, if it exists:
$$\lim_{x\to\infty } \left(\sqrt{e^{2x}+e^{x}}-\sqrt{e^{2x}-e^{x}}\right)$$

REPLY [13 votes]: Another way could be to consider $$A=\sqrt{e^{2x}+e^{x}}-\sqrt{e^{2x}-e^{x}}=e^x\Big(\sqrt{1+\frac 1 {e^x}}-\sqrt{1-\frac 1 {e^x}}\Big)$$ and use Taylor series (or the generalized binomial theorem) $$\sqrt{1+y}=1+\frac{y}{2}-\frac{y^2}{8}+O\left(y^3\right)$$ Replacing $y$ by $e^{-x}$ in the first term and by $-e^{-x}$ in the second term lead to $$A=1+\frac{e^{-2 x}}{8}+\cdots$$ which shows the limit and how it is approached.