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

Explanation:

Step1: Rationalize the numerator

Multiply by $\frac{\sqrt{3x + 16}+5}{\sqrt{3x + 16}+5}$. [ \begin{align*} &\lim_{x\rightarrow3}\frac{\sqrt{3x + 16}-5}{x - 3}\times\frac{\sqrt{3x + 16}+5}{\sqrt{3x + 16}+5}\ =&\lim_{x\rightarrow3}\frac{(3x + 16)-25}{(x - 3)(\sqrt{3x + 16}+5)}\ =&\lim_{x\rightarrow3}\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\rightarrow3}\frac{3(x - 3)}{(x - 3)(\sqrt{3x + 16}+5)}\ =&\lim_{x\rightarrow3}\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:

$\frac{3}{10}$