Solve for: \lim_{x arrow-sqrt(5)} ((x^2-2sqrt(5)x-15)/(x+sqrt(5)))

Expression: $\lim_{x \rightarrow -\sqrt{ 5 }} \left(\frac{ {x}^{2}-2\sqrt{ 5 }x-15 }{ x+\sqrt{ 5 } }\right)$

Since evaluating limits of the numerator and denominator would result in an indeterminate form, use the L'Hopital's rule

$\lim_{x \rightarrow -\sqrt{ 5 }} \left(\frac{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( {x}^{2}-2\sqrt{ 5 }x-15 \right) }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( x+\sqrt{ 5 } \right) }\right)$

Find the derivative

$\lim_{x \rightarrow -\sqrt{ 5 }} \left(\frac{ 2x-2\sqrt{ 5 } }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( x+\sqrt{ 5 } \right) }\right)$

Find the derivative

$\lim_{x \rightarrow -\sqrt{ 5 }} \left(\frac{ 2x-2\sqrt{ 5 } }{ 1 }\right)$

Any expression divided by $1$ remains the same

$\lim_{x \rightarrow -\sqrt{ 5 }} \left(2x-2\sqrt{ 5 }\right)$

Evaluate the limit

$2 \times \left( -\sqrt{ 5 } \right)-2\sqrt{ 5 }$

Simplify the expression

$\begin{align*}&-4\sqrt{ 5 } \\&\approx-8.94427\end{align*}$