Question

Let $f(x)=\frac{\sqrt{4+x}-2}{x},x\ne 0$. 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)

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