# How do you find the limit lim_(x->9)(9-x)/(3-sqrt(x)) ?

Sep 7, 2014

By cancelling common factors, we can find
${\lim}_{x \to 9} \frac{9 - x}{3 - \sqrt{x}} = 6$.

Let us look at some details.
The first thing we should try when evaluating a limit is plug in the value. In this posted limit, we get $\frac{0}{0}$ when we plug in $x = 9$, which indicates that there should be a common factor $\left(9 - x\right)$ hidden in the expression. Since the factor $\left(9 - x\right)$ is already visible in the numerator, let us squeeze the factor out of the denominator.

By multiplying the numerator and the denominator by $3 + \sqrt{x}$ ,
lim_{x to 9}{9-x}/{3-sqrt{x}}cdot{3+sqrt{x}}/{3+sqrt{x}} =lim_{x to 9}{(9-x)(3+sqrt{x})}/{9-x}
by cancelling out $\left(9 - x\right)$,
$= {\lim}_{x \to 9} \left(3 + \sqrt{x}\right) = 3 + \sqrt{9} = 6$