evaluate the limit: $$\\lim_{x \\to -5} \\frac{\\sqrt{x + 14} - 8}{-2x - 2}$$
evaluate the limit: $$\\lim_{x \\to -5} \\frac{\\sqrt{x + 14} - 8}{-2x - 2}$$
Answer
Explanation:
Step1: Substitute (x = - 5) into the function
Substituting (x=-5) into (\frac{\sqrt{x + 14}-8}{-2x - 2}) gives (\frac{\sqrt{-5 + 14}-8}{-2\times(-5)-2}=\frac{\sqrt{9}-8}{10 - 2}=\frac{3 - 8}{8}=\frac{-5}{8}). But we can also use the rational - ization method. 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}&=\lim_{x\rightarrow - 5}\frac{(\sqrt{x + 14}-8)(\sqrt{x + 14}+8)}{(-2x - 2)(\sqrt{x + 14}+8)}\ \end{align*} ]
Step2: Simplify the numerator using the difference of squares formula ((a - b)(a + b)=a^{2}-b^{2})
The numerator ((\sqrt{x + 14}-8)(\sqrt{x + 14}+8)=(x + 14)-64=x-50) The denominator ((-2x - 2)(\sqrt{x + 14}+8)=-2(x + 1)(\sqrt{x + 14}+8)) So the limit becomes (\lim_{x\rightarrow - 5}\frac{x-50}{-2(x + 1)(\sqrt{x + 14}+8)})
Step3: Substitute (x=-5) into the simplified function
Substitute (x = - 5) into (\frac{x-50}{-2(x + 1)(\sqrt{x + 14}+8)}) [ \begin{align*} \frac{-5-50}{-2(-5 + 1)(\sqrt{-5+14}+8)}&=\frac{-55}{-2\times(-4)\times(3 + 8)}\ &=\frac{-55}{8\times11}\ &=-\frac{5}{8} \end{align*} ]
Answer:
(-\frac{5}{8})