ii) $lim_{x\rightarrow4}\frac{2 - sqrt{x}}{x - 4}$
ii) $lim_{x\rightarrow4}\frac{2 - sqrt{x}}{x - 4}$
Answer
Explanation:
Step1: Rationalize the numerator
Multiply the fraction by $\frac{2 + \sqrt{x}}{2+\sqrt{x}}$. [ \begin{align*} \lim_{x\rightarrow4}\frac{2-\sqrt{x}}{x - 4}\times\frac{2+\sqrt{x}}{2+\sqrt{x}}&=\lim_{x\rightarrow4}\frac{4 - x}{(x - 4)(2+\sqrt{x})}\ &=\lim_{x\rightarrow4}\frac{-(x - 4)}{(x - 4)(2+\sqrt{x})} \end{align*} ]
Step2: Simplify the fraction
Cancel out the common factor $(x - 4)$ (since $x\neq4$ when taking the limit). [ \begin{align*} \lim_{x\rightarrow4}\frac{-(x - 4)}{(x - 4)(2+\sqrt{x})}&=\lim_{x\rightarrow4}\frac{- 1}{2+\sqrt{x}} \end{align*} ]
Step3: Substitute $x = 4$
[ \begin{align*} \lim_{x\rightarrow4}\frac{-1}{2+\sqrt{x}}&=\frac{-1}{2+\sqrt{4}}\ &=\frac{-1}{2 + 2}\ &=-\frac{1}{4} \end{align*} ]
Answer:
$-\frac{1}{4}$