Question

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

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:

$\frac{3}{10}$