evaluate:
lim x approach infinity [sqrt(e^2x-e^x)-e^x]
tried alotta things, didn work, help pls!
$\displaystyle = \lim_{x \rightarrow + \infty}\frac{\sqrt{(e^{2x} - e^x} - e^x)(\sqrt{e^{2x} - e^x} + e^x)}{\sqrt{e^{2x} - e^x} + e^x}$
$\displaystyle = \lim_{x \rightarrow + \infty}\frac{e^{2x} - e^x - e^{2x}}{\sqrt{e^{2x} - e^x} + e^x}$
$\displaystyle = \lim_{x \rightarrow + \infty} - \frac{e^x}{\sqrt{e^{2x} - e^x} + e^x}$
$\displaystyle = \lim_{x \rightarrow + \infty} - \frac{1}{\sqrt{1 - e^{-x}} + 1}$
$\displaystyle = - \frac{1}{2}$.
I divided the numerator and denominator by $\displaystyle e^x$. When you divide the numerator by $\displaystyle e^x$ you get 1.
Note on denominator: When dividing outside the square root by $\displaystyle e^x$, you divide inside the square root by $\displaystyle e^{2x}$:
$\displaystyle \frac{\sqrt{e^{2x} - e^x}}{e^x} = \sqrt{\frac{e^{2x} - e^x}{e^{2x}}} = \sqrt{1 - e^{-x}}$.