Question

ii) $lim_{x
ightarrow4}\frac{2 - sqrt{x}}{x - 4}$

Explanation:

Step1: Rationalize the numerator

Multiply the fraction by $\frac{2 + \sqrt{x}}{2+\sqrt{x}}$.
\[

$$\begin{align*} \lim_{x ightarrow4}\frac{2-\sqrt{x}}{x - 4}\times\frac{2+\sqrt{x}}{2+\sqrt{x}}&=\lim_{x ightarrow4}\frac{4 - x}{(x - 4)(2+\sqrt{x})}\\ &=\lim_{x ightarrow4}\frac{-(x - 4)}{(x - 4)(2+\sqrt{x})} \end{align*}$$

\]

Step2: Simplify the fraction

Cancel out the common factor $(x - 4)$ (since $x
eq4$ when taking the limit).
\[

$$\begin{align*} \lim_{x ightarrow4}\frac{-(x - 4)}{(x - 4)(2+\sqrt{x})}&=\lim_{x ightarrow4}\frac{- 1}{2+\sqrt{x}} \end{align*}$$

\]

Step3: Substitute $x = 4$

\[

$$\begin{align*} \lim_{x ightarrow4}\frac{-1}{2+\sqrt{x}}&=\frac{-1}{2+\sqrt{4}}\\ &=\frac{-1}{2 + 2}\\ &=-\frac{1}{4} \end{align*}$$

\]

Answer:

$-\frac{1}{4}$