# Question #061a3

Dec 27, 2017

$- \frac{5}{6}$

#### Explanation:

${\lim}_{x \rightarrow 2} \left(\frac{\sqrt{{x}^{2} + 5} - 4 x + 5}{{x}^{2} - 4}\right)$

${\lim}_{x \rightarrow 2} \left(\frac{\left(\sqrt{{x}^{2} + 5} - \left(4 x - 5\right)\right) \left(\sqrt{{x}^{2} + 5} + \left(4 x - 5\right)\right)}{\left({x}^{2} - 4\right) \left(\sqrt{{x}^{2} + 5} + \left(4 x - 5\right)\right)}\right)$ (conjugate both sides)

${\lim}_{x \rightarrow 2} \left(\frac{{x}^{2} + 5 - {\left(4 x - 5\right)}^{2}}{\left({x}^{2} - 4\right) \left(\sqrt{{x}^{2} + 5} + \left(4 x - 5\right)\right)}\right)$

${\lim}_{x \rightarrow 2} \left(\frac{{x}^{2} + 5 - 16 {x}^{2} + 40 x - 25}{\left({x}^{2} - 4\right) \left(\sqrt{{x}^{2} + 5} + \left(4 x - 5\right)\right)}\right)$

${\lim}_{x \rightarrow 2} \left(\frac{- 15 {x}^{2} + 40 x - 20}{\left({x}^{2} - 4\right) \left(\sqrt{{x}^{2} + 5} + \left(4 x - 5\right)\right)}\right)$

${\lim}_{x \rightarrow 2} \left(\frac{- 5 \left(3 {x}^{2} - 8 x + 4\right)}{\left({x}^{2} - 4\right) \left(\sqrt{{x}^{2} + 5} + \left(4 x - 5\right)\right)}\right)$

${\lim}_{x \rightarrow 2} \left(\frac{- 5 \left(3 x - 2\right) \left(x - 2\right)}{\left(x + 2\right) \left(x - 2\right) \left(\sqrt{{x}^{2} + 5} + \left(4 x - 5\right)\right)}\right)$

${\lim}_{x \rightarrow 2} \left(\frac{- 5 \left(3 x - 2\right)}{\left(x + 2\right) \left(\sqrt{{x}^{2} + 5} + \left(4 x - 5\right)\right)}\right)$ (factor out x-2)

direct substitution: $= \frac{- 5 \left(3 \left(2\right) - 2\right)}{\left(2 + 2\right) \left(\sqrt{{2}^{2} + 5} + \left(4 \left(2\right) - 5\right)\right)}$

$= \frac{- 5 \left(4\right)}{4 \left(\sqrt{9} + 3\right)}$

$= - \frac{20}{4 \left(6\right)}$

$= - \frac{5}{6}$