# What is the limit of (2-sqrt(x))/(4-x) as x approaches 4?

A direct substitution results in the indeterminate form of $\frac{0}{0}$ so we resort to L'hospital rule which states that we take the derivative of the numerator and then the denominator and then attempt to apply the limit again.
derivative of the numerator = $- \frac{1}{2 \sqrt{x}}$
${\lim}_{x \to 4} \frac{2 - \sqrt{x}}{4 - x} = {\lim}_{x \to 4} \frac{- \frac{1}{2 \sqrt{x}}}{- 1} = {\lim}_{x \to 4} \frac{1}{2 \sqrt{x}} = \frac{1}{2 \sqrt{4}} = \frac{1}{2 \cdot 2} = \frac{1}{4} = 0.25$