$$\\lim_{x \\to 11} \\frac{\\sqrt{x - 7} - 2}{-x + 11}$$
$$\\lim_{x \\to 11} \\frac{\\sqrt{x - 7} - 2}{-x + 11}$$
Answer
Explanation:
Step1: Rationalize the numerator
Multiply the numerator and denominator by $\sqrt{x - 7}+2$. $$\lim_{x\rightarrow11}\frac{(\sqrt{x - 7}-2)(\sqrt{x - 7}+2)}{(-x + 11)(\sqrt{x - 7}+2)}$$ 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 $\lim_{x\rightarrow11}\frac{x - 11}{(-x + 11)(\sqrt{x - 7}+2)}$.
Step2: Simplify the expression
Factor out $-1$ from the denominator: $-x + 11=-(x - 11)$. The limit becomes $\lim_{x\rightarrow11}\frac{x - 11}{-(x - 11)(\sqrt{x - 7}+2)}$. Cancel out the $(x - 11)$ terms (since $x\neq11$ as we are taking the limit as $x\rightarrow11$), we get $\lim_{x\rightarrow11}\frac{1}{-(\sqrt{x - 7}+2)}$.
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:
$-\frac{1}{4}$