Question

Examine the continuity of the function:
$f\left(x\right)=\frac{\log 100+\log (0.01+x)}{3x}$, for $x\ne 0$
$=\frac{100}{3}$ $forx=0;atx=0$

Answer 1

$f\left(x\right)=\frac{\log 100+\log (0.01+x)}{3x}$

$\underset{x\to 0}{lim}f\left(x\right)=\frac{\log \left(100\right)+\log (0.01+x)}{0}\frac{0}{0}$form

$=\underset{x\to 0}{lim}\frac{0+\left(\frac{1}{0.01+x}\right).1}{3}$

$=\frac{1}{0.01}\times \frac{1}{3}=\frac{100}{3}$

$f\left(0\right)$ is given as $\frac{100}{3}$

$\Rightarrow \underset{x\to 0}{lim}f\left(x\right)=f\left(0\right)$

$\Rightarrow$It is continuous at $x=0$