Evaluating a Limit Containing a Root Using a Conjugate

$\begin{array}{ll}\underset{x\to 0}{\lim }\left(\frac{\sqrt{25-x}-5}{x}\right)=\underset{x\to 0}{\lim }\left(\frac{\left(\sqrt{25-x}-5\right)}{x}\cdot \frac{\left(\sqrt{25-x}+5\right)}{\left(\sqrt{25-x}+5\right)}\right)\hfill & \text{Multiply numerator and denominator by the conjugate}.\hfill \\ =\underset{x\to 0}{\lim }\left(\frac{\left(25-x\right)-25}{x\left(\sqrt{25-x}+5\right)}\right)\hfill & \text{Multiply: }\left(\sqrt{25-x}-5\right)\cdot \left(\sqrt{25-x}+5\right)=\left(25-x\right)-25.\hfill \\ =\underset{x\to 0}{\lim }\left(\frac{-x}{x\left(\sqrt{25-x}+5\right)}\right)\hfill & \text{Combine like terms}.\hfill \\ =\underset{x\to 0}{\lim }\left(\frac{-\cancel{x}}{\cancel{x}\left(\sqrt{25-x}+5\right)}\right)\hfill & \text{Simplify }\frac{-x}{x}=-1.\hfill \\ =\frac{-1}{\sqrt{25-0}+5}\hfill & \text{Evaluate}.\hfill \\ =\frac{-1}{5+5}=-\frac{1}{10}\hfill & \hfill \end{array}$