Question

find the limit.
lim_{x
ightarrow81}\frac{sqrt{x}-9}{x - 81}
select the correct choice below and, if necessary, fill in the blank.
a. (lim_{x
ightarrow81}\frac{sqrt{x}-9}{x - 81}=) (type an integer or a simplified fraction.)
b. the limit does not exist.

Explanation:

Step1: Rationalize the numerator

Multiply the fraction $\frac{\sqrt{x}-9}{x - 81}$ by $\frac{\sqrt{x}+9}{\sqrt{x}+9}$. We get $\frac{(\sqrt{x}-9)(\sqrt{x}+9)}{(x - 81)(\sqrt{x}+9)}$. Using the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$, the numerator becomes $x-81$. So the fraction is $\frac{x - 81}{(x - 81)(\sqrt{x}+9)}$.

Step2: Simplify the fraction

Cancel out the common factor $(x - 81)$ in the numerator and the denominator. The simplified fraction is $\frac{1}{\sqrt{x}+9}$.

Step3: Evaluate the limit

Substitute $x = 81$ into $\frac{1}{\sqrt{x}+9}$. We have $\frac{1}{\sqrt{81}+9}=\frac{1}{9 + 9}=\frac{1}{18}$.

Answer:

A. $\lim_{x
ightarrow81}\frac{\sqrt{x}-9}{x - 81}=\frac{1}{18}$