# Evaluate Limit $\lim_{\epsilon\to 0}\frac{1}{\epsilon}\left(\frac{1}{\sqrt{4 + \epsilon}} - \frac{1}{2}\right)$

$$\lim_{\epsilon\to 0}\frac{1}{\epsilon}\left(\frac{1}{\sqrt{4 + \epsilon}} - \frac{1}{2}\right)$$

I know the limit is $-\frac{1}{16}$, but I just can't figure out how comes... Can anyone help me with a hint?

HINT : Yours will be $$\frac 1\epsilon\times\frac{2-\sqrt{4+\epsilon}}{2\sqrt{4+\epsilon}}=\frac 1\epsilon \times\frac{4-(4+\epsilon)}{2\sqrt{4+\epsilon}\times \left(2+\sqrt{4+\epsilon}\right)}=\frac{-1}{2\sqrt{4+\epsilon}\times \left(2+\sqrt{4+\epsilon}\right)}$$

• You are welcome! – mathlove Jan 6 '14 at 17:32