Question

$$\lim_{x \to 11} \frac{\sqrt{x - 7} - 2}{-x + 11}$$

Explanation:

Step1: Rationalize the numerator

Multiply the numerator and denominator by $\sqrt{x - 7}+2$.

$$\lim_{x ightarrow11}\frac{(\sqrt{x - 7}-2)(\sqrt{x - 7}+2)}{(-x + 11)(\sqrt{x - 7}+2)}$$

Using the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$, the numerator becomes $(x - 7)-4=x - 11$.
So the limit is $\lim_{x
ightarrow11}\frac{x - 11}{(-x + 11)(\sqrt{x - 7}+2)}$.

Step2: Simplify the expression

Factor out $-1$ from the denominator: $-x + 11=-(x - 11)$.
The limit becomes $\lim_{x
ightarrow11}\frac{x - 11}{-(x - 11)(\sqrt{x - 7}+2)}$.
Cancel out the $(x - 11)$ terms (for $x
eq11$, as we are taking the limit as $x
ightarrow11$), we get $\lim_{x
ightarrow11}\frac{1}{-(\sqrt{x - 7}+2)}$.

Step3: Substitute $x = 11$

Substitute $x = 11$ into the simplified expression: $\frac{1}{-(\sqrt{11 - 7}+2)}$.
Since $\sqrt{11 - 7}=\sqrt{4}=2$, then $\frac{1}{-(2 + 2)}=-\frac{1}{4}$.

Answer:

$-\frac{1}{4}$