question evaluate the limit: $lim_{x\rightarrow9}\frac{32 - 2x}{4-sqrt{x}}$
question evaluate the limit: $lim_{x\rightarrow9}\frac{32 - 2x}{4-sqrt{x}}$
Answer
Answer:
16
Explanation:
Step1: Rationalize the denominator
Multiply numerator and denominator by $4 + \sqrt{x}$: [ \begin{align*} &\lim_{x\rightarrow9}\frac{(32 - 2x)(4+\sqrt{x})}{(4-\sqrt{x})(4 + \sqrt{x})}\ =&\lim_{x\rightarrow9}\frac{(32 - 2x)(4+\sqrt{x})}{16 - x} \end{align*} ]
Step2: Factor out 2 from the numerator
Factor out 2 from $32 - 2x$ to get $2(16 - x)$: [ \begin{align*} &\lim_{x\rightarrow9}\frac{2(16 - x)(4+\sqrt{x})}{16 - x}\ =&\lim_{x\rightarrow9}2(4+\sqrt{x}) \end{align*} ]
Step3: Substitute $x = 9$
[ \begin{align*} &2(4+\sqrt{9})\ =&2(4 + 3)\ =&2\times7\ =&16 \end{align*} ]