Question

find the limit.
lim_{x
ightarrow100}\frac{sqrt{x}-10}{x - 100}
select the correct choice below and, if necessary, fill in the answer box to co
a. (lim_{x
ightarrow100}\frac{sqrt{x}-10}{x - 100}=\frac{1}{20}) (type an integer or a simplified fraction )
b. the limit does not exist.

Explanation:

Step1: Rationalize the numerator

Multiply the fraction $\frac{\sqrt{x}-10}{x - 100}$ by $\frac{\sqrt{x}+10}{\sqrt{x}+10}$. We get $\frac{(\sqrt{x}-10)(\sqrt{x}+10)}{(x - 100)(\sqrt{x}+10)}$. Using the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$, the numerator becomes $x - 100$. So the fraction is $\frac{x - 100}{(x - 100)(\sqrt{x}+10)}$.

Step2: Simplify the fraction

Cancel out the common factor $(x - 100)$ in the numerator and the denominator. The simplified fraction is $\frac{1}{\sqrt{x}+10}$.

Step3: Evaluate the limit

Substitute $x = 100$ into $\frac{1}{\sqrt{x}+10}$. We have $\frac{1}{\sqrt{100}+10}=\frac{1}{10 + 10}=\frac{1}{20}$.

Answer:

A. $\lim_{x
ightarrow100}\frac{\sqrt{x}-10}{x - 100}=\frac{1}{20}$