question\nevaluate the limit: $$ lim _ { x \rightarrow - 7 } \frac { sqrt { x + 11 } - 2 } { 2 x + 14 } $$
question\nevaluate the limit: $$ lim _ { x \rightarrow - 7 } \frac { sqrt { x + 11 } - 2 } { 2 x + 14 } $$
Answer
Explanation:
Step1: Rationalize the numerator
Multiply the numerator and denominator by $\sqrt{x + 11}+2$ $$\lim_{x\rightarrow - 7}\frac{(\sqrt{x + 11}-2)(\sqrt{x + 11}+2)}{(2x + 14)(\sqrt{x + 11}+2)}$$ Using the formula $(a - b)(a + b)=a^{2}-b^{2}$, the numerator becomes $(x + 11)-4=x + 7$ The denominator: $2x+14 = 2(x + 7)$ So the limit becomes $$\lim_{x\rightarrow - 7}\frac{x + 7}{2(x + 7)(\sqrt{x + 11}+2)}$$
Step2: Simplify the expression
Cancel out the common factor $(x + 7)$ (since $x\neq - 7$ when taking the limit, $x\rightarrow - 7$ but $x\neq - 7$) We get $$\lim_{x\rightarrow - 7}\frac{1}{2(\sqrt{x + 11}+2)}$$
Step3: Substitute $x=-7$
Substitute $x =-7$ into the simplified expression: $$\frac{1}{2(\sqrt{-7 + 11}+2)}=\frac{1}{2(2 + 2)}$$
Answer:
$\frac{1}{8}$