t: \\( \\lim _ { x \\rightarrow - 5 } \\frac { \\sqrt { x + 14 } - 8 } { - 2 x - 2 } \\)

Explanation:

Step1: Rationalize the numerator

Multiply the numerator and denominator by the conjugate of the numerator (\sqrt{x + 14}+8) [ \begin{align*} &\lim_{x\rightarrow - 5}\frac{\sqrt{x + 14}-8}{-2x - 2}\times\frac{\sqrt{x + 14}+8}{\sqrt{x + 14}+8}\ =&\lim_{x\rightarrow - 5}\frac{(x + 14)-64}{(-2x - 2)(\sqrt{x + 14}+8)}\ =&\lim_{x\rightarrow - 5}\frac{x - 50}{(-2x - 2)(\sqrt{x + 14}+8)} \end{align*} ]

Step2: Substitute (x=-5)

Substitute (x =-5) into the expression (\frac{x - 50}{(-2x - 2)(\sqrt{x + 14}+8)}) [ \begin{align*} &\frac{-5-50}{(-2\times(-5)-2)(\sqrt{-5 + 14}+8)}\ =&\frac{-55}{(10 - 2)(3 + 8)}\ =&\frac{-55}{8\times11}\ =&-\frac{5}{8} \end{align*} ]

Answer:

(-\frac{5}{8})