use properties of limits to find the indicated limit. it may be necessary to rewrite an expression before…

use properties of limits to find the indicated limit. it may be necessary to rewrite an expression before limit properties can be applied. lim(x→16) (√x - 4)/(x - 16) select the correct choice below and fill in any answer boxes in your choice. a. lim(x→16) (√x - 4)/(x - 16) = (type an integer or a simplified fraction.) b. the limit does not exist.

use properties of limits to find the indicated limit. it may be necessary to rewrite an expression before limit properties can be applied. lim(x→16) (√x - 4)/(x - 16) select the correct choice below and fill in any answer boxes in your choice. a. lim(x→16) (√x - 4)/(x - 16) = (type an integer or a simplified fraction.) b. the limit does not exist.

Answer

Explanation:

Step1: Rationalize the numerator

Multiply the fraction $\frac{\sqrt{x}-4}{x - 16}$ by $\frac{\sqrt{x}+4}{\sqrt{x}+4}$. We get $\frac{(\sqrt{x}-4)(\sqrt{x}+4)}{(x - 16)(\sqrt{x}+4)}$. Using the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$, the numerator becomes $x-16$. So the fraction is $\frac{x - 16}{(x - 16)(\sqrt{x}+4)}$.

Step2: Simplify the fraction

Cancel out the common factor $(x - 16)$ in the numerator and denominator (since $x\neq16$ when taking the limit). The simplified fraction is $\frac{1}{\sqrt{x}+4}$.

Step3: Apply the limit

Now, find $\lim_{x\rightarrow16}\frac{1}{\sqrt{x}+4}$. Substitute $x = 16$ into the function $\frac{1}{\sqrt{x}+4}$. We have $\frac{1}{\sqrt{16}+4}=\frac{1}{4 + 4}=\frac{1}{8}$.

Answer:

A. $\lim_{x\rightarrow16}\frac{\sqrt{x}-4}{x - 16}=\frac{1}{8}$