Question

find the following limit or state that it does not exist
\\( \lim\limits_{x\to 2} \frac{x - 2}{\sqrt{4x + 1} - 3}

simplify the given limit.
\\( \lim\limits_{x\to 2} \frac{x - 2}{\sqrt{4x + 1} - 3} = \lim\limits_{x\to 2} \left( \frac{\sqrt{4x + 1} + 3}{4} \
ight) \\) (simplify your answer)

evaluate the limit, if possible. select the correct choice below and, if necessary, fill in the answer box to complete your choice
\\( \bigcirc \\) a. \\( \lim\limits_{x\to 2} \frac{x - 2}{\sqrt{4x + 1} - 3} = \square \\) (type an exact answer.)
\\( \bigcirc \\) b. the limit does not exist.

Explanation:

Step1: Substitute the simplified form

We have the simplified limit as $\lim\limits_{x\to 2}\frac{\sqrt{4x + 1}+3}{4}$. Now we substitute $x = 2$ into the expression.

Step2: Calculate the value

First, calculate the value inside the square root: $4\times2+ 1=9$. Then $\sqrt{9}=3$. So the numerator becomes $3 + 3=6$. Then we have $\frac{6}{4}=\frac{3}{2}$.

Answer:

A. $\lim\limits_{x\to 2}\frac{x - 2}{\sqrt{4x + 1}-3}=\frac{3}{2}$