Question

Can you help me solving a limit

Answer 1

\[\begin{align*}\lim_{x\to2}\frac{\dfrac{1}{\sqrt{2+x}}-\dfrac{1}{2}}{x-2}&=\lim_{x\to2}\frac{\dfrac{2}{2\sqrt{2+x}}-\dfrac{\sqrt{2+x}}{2\sqrt{2+x}}}{x-2}\\\\
&=\lim_{x\to2}\frac{\dfrac{2-\sqrt{2+x}}{2\sqrt{2+x}}}{x-2}\\\\
&=\lim_{x\to2}\frac{\dfrac{2-\sqrt{2+x}}{2\sqrt{2+x}}}{x-2}\cdot\frac{2+\sqrt{2+x}}{2+\sqrt{2+x}}\\\\
&=\lim_{x\to2}\frac{\dfrac{4-(2+x)}{2\sqrt{2+x}}}{(x-2)(2+\sqrt{2+x})}\\\\
&=\lim_{x\to2}\frac{\dfrac{2-x}{2\sqrt{2+x}}}{(x-2)(2+\sqrt{2+x})}\\\\
&=-\lim_{x\to2}\frac{\dfrac{1}{2\sqrt{2+x}}}{2+\sqrt{2+x}}\end{align*}\]

Answer 2

Thank you so much!.

Answer 3

yw