evaluate the limit. if the limit does not exist, enter dne. lim_{x→16} (x - 16)/(√x - 4)

Explanation:

Step1: Rationalize the denominator

Multiply numerator and denominator by $\sqrt{x}+4$. [ \begin{align*} \lim_{x\rightarrow16}\frac{x - 16}{\sqrt{x}-4}&=\lim_{x\rightarrow16}\frac{(x - 16)(\sqrt{x}+4)}{(\sqrt{x}-4)(\sqrt{x}+4)}\ \end{align*} ]

Step2: Simplify the denominator

Use the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$. Here $a=\sqrt{x}$ and $b = 4$, so $(\sqrt{x}-4)(\sqrt{x}+4)=x - 16$. [ \begin{align*} \lim_{x\rightarrow16}\frac{(x - 16)(\sqrt{x}+4)}{x - 16}&=\lim_{x\rightarrow16}(\sqrt{x}+4) \end{align*} ]

Step3: Evaluate the limit

Substitute $x = 16$ into $\sqrt{x}+4$. [ \sqrt{16}+4=4 + 4=8 ]

Answer:

$8$