evaluate the limit: $lim_{x
ightarrow0}\frac{sqrt{2x + 9}-3}{x}$

Question

Accepted Answer

# Explanation:
## Step1: Rationalize the numerator
Multiply the fraction by $\frac{\sqrt{2x + 9}+3}{\sqrt{2x + 9}+3}$.
$$
\begin{align*}
&\lim_{x
ightarrow0}\frac{\sqrt{2x + 9}-3}{x}\times\frac{\sqrt{2x + 9}+3}{\sqrt{2x + 9}+3}\
=&\lim_{x
ightarrow0}\frac{(\sqrt{2x + 9})^2-3^2}{x(\sqrt{2x + 9}+3)}\
=&\lim_{x
ightarrow0}\frac{2x + 9 - 9}{x(\sqrt{2x + 9}+3)}\
=&\lim_{x
ightarrow0}\frac{2x}{x(\sqrt{2x + 9}+3)}
\end{align*}
$$
## Step2: Simplify the fraction
Cancel out the common - factor $x$ in the numerator and denominator.
$$
\begin{align*}
&\lim_{x
ightarrow0}\frac{2x}{x(\sqrt{2x + 9}+3)}\
=&\lim_{x
ightarrow0}\frac{2}{\sqrt{2x + 9}+3}
\end{align*}
$$
## Step3: Evaluate the limit
Substitute $x = 0$ into the simplified function.
$$
\begin{align*}
&\frac{2}{\sqrt{2\times0+9}+3}\
=&\frac{2}{\sqrt{9}+3}\
=&\frac{2}{3 + 3}\
=&\frac{1}{3}
\end{align*}
$$
# Answer:
$\frac{1}{3}$

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QUESTION IMAGE

Question

Explanation:

Step1: Rationalize the numerator

Step2: Simplify the fraction

Step3: Evaluate the limit

Answer: