4. evaluate\n lim_{x\rightarrow16}\frac{16 - x}{4-sqrt{x}}
4. evaluate\n lim_{x\rightarrow16}\frac{16 - x}{4-sqrt{x}}
Answer
Explanation:
Step1: Rationalize the denominator
Multiply numerator and denominator by $4 + \sqrt{x}$. [ \begin{align*} \lim_{x\rightarrow16}\frac{16 - x}{4-\sqrt{x}}\times\frac{4+\sqrt{x}}{4+\sqrt{x}}&=\lim_{x\rightarrow16}\frac{(16 - x)(4+\sqrt{x})}{16 - x} \end{align*} ]
Step2: Simplify the expression
Cancel out the common factor $(16 - x)$ (since $x\neq16$ when taking the limit). [ \begin{align*} \lim_{x\rightarrow16}\frac{(16 - x)(4+\sqrt{x})}{16 - x}&=\lim_{x\rightarrow16}(4+\sqrt{x}) \end{align*} ]
Step3: Substitute the value of $x$
Substitute $x = 16$ into $4+\sqrt{x}$. [ \begin{align*} 4+\sqrt{16}&=4 + 4=8 \end{align*} ]
Answer:
$8$