Question

Let $f(x)=\frac{\sqrt{4+x}-2}{x}, x\neq 0$. For $f(x)$ to be continuous at $x$ = 0, we must have $f(0)$ is equal to

• A. 0
• B. 4
• C. $\frac{1}{4}$
• D. 1

Answer 1

Answer: (C)

$\lim_{x\to0} f\left(x\right) = \lim _{x\to 0} \frac{\sqrt{4+x} -2}{x}$
= $\lim _{x\to 0} \frac{\sqrt{4+x} -2}{x} . \frac{\sqrt{4+ x} +2}{\sqrt{4+x} +2} $
= $\lim _{x\to 0} \frac{4+x -4}{x\left[\sqrt{4+x} +2\right]} $
= $ \lim _{x\to 0} \frac{1}{\sqrt{4+x }+2} = \frac{1}{2+2} = \frac{1}{4} $
hence $f\left(0\right)= \frac{1}{4}$