Question

find the following limit or state that it does not exist.
(limlimits_{h\to 0} \frac{9}{sqrt{64 + 9h} + 2})

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
(\bigcirc) a. (limlimits_{h\to 0} \frac{9}{sqrt{64 + 9h} + 2} = square) (simplify your answer)
(\bigcirc) b. the limit does not exist.

Explanation:

Step1: Substitute h = 0 into the function

To find the limit as \( h \to 0 \), we substitute \( h = 0 \) into the expression \( \frac{9}{\sqrt{64 + 9h}+2} \).
\[
\frac{9}{\sqrt{64 + 9(0)}+2}
\]

Step2: Simplify the square root and denominator

First, simplify the square root: \( \sqrt{64 + 0} = \sqrt{64} = 8 \). Then the denominator becomes \( 8 + 2 = 10 \). So the expression is \( \frac{9}{10} \).

Answer:

\(\frac{9}{10}\) (and the correct choice is A)