Question

question
evaluate the limit:

$$ lim _ { x ightarrow - 7 } \frac { sqrt { x + 11 } - 2 } { 2 x + 14 } $$

Explanation:

Step1: Rationalize the numerator

Multiply the numerator and denominator by $\sqrt{x + 11}+2$

$$\lim_{x ightarrow - 7}\frac{(\sqrt{x + 11}-2)(\sqrt{x + 11}+2)}{(2x + 14)(\sqrt{x + 11}+2)}$$

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

$$\lim_{x ightarrow - 7}\frac{x + 7}{2(x + 7)(\sqrt{x + 11}+2)}$$

Step2: Simplify the expression

Cancel out the common factor $(x + 7)$ (since $x
eq - 7$ when taking the limit, $x
ightarrow - 7$ but $x
eq - 7$)
We get

$$\lim_{x ightarrow - 7}\frac{1}{2(\sqrt{x + 11}+2)}$$

Step3: Substitute $x=-7$

Substitute $x =-7$ into the simplified expression:
$$\frac{1}{2(\sqrt{-7 + 11}+2)}=\frac{1}{2(2 + 2)}$$

Answer:

$\frac{1}{8}$