16. $lim_{x o 3}\frac{sqrt{3x + 16}-5}{x - 3}$

Question

16. $lim_{x 	o 3}\frac{sqrt{3x + 16}-5}{x - 3}$

Accepted Answer

# Explanation:
## Step1: Rationalize the numerator
Multiply by $\frac{\sqrt{3x + 16}+5}{\sqrt{3x + 16}+5}$.
$$
\begin{align*}
&\lim_{x
ightarrow3}\frac{\sqrt{3x + 16}-5}{x - 3}\times\frac{\sqrt{3x + 16}+5}{\sqrt{3x + 16}+5}\
=&\lim_{x
ightarrow3}\frac{(3x + 16)-25}{(x - 3)(\sqrt{3x + 16}+5)}\
=&\lim_{x
ightarrow3}\frac{3x-9}{(x - 3)(\sqrt{3x + 16}+5)}
\end{align*}
$$
## Step2: Simplify the fraction
Factor out 3 from the numerator.
$$
\begin{align*}
&\lim_{x
ightarrow3}\frac{3(x - 3)}{(x - 3)(\sqrt{3x + 16}+5)}\
=&\lim_{x
ightarrow3}\frac{3}{\sqrt{3x + 16}+5}
\end{align*}
$$
## Step3: Substitute $x = 3$
$$
\begin{align*}
&\frac{3}{\sqrt{3\times3+16}+5}\
=&\frac{3}{\sqrt{9 + 16}+5}\
=&\frac{3}{\sqrt{25}+5}\
=&\frac{3}{5 + 5}\
=&\frac{3}{10}
\end{align*}
$$
# Answer:
$\frac{3}{10}$

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QUESTION IMAGE

Question

Explanation:

Step1: Rationalize the numerator

Step2: Simplify the fraction

Step3: Substitute $x = 3$

Answer: