Question

If $f\left(x\right)=\{\begin{array}{ccc}\frac{x}{e^{\frac{1}{x}}+1},& when& x\ne 0\\ 0,& when& x=0\end{array}$ then

• A. ${lim}_{x\to 0+}f\left(x\right)=1$
• B. ${lim}_{x\to 0-}f\left(x\right)=1$
• C. f(x) is continuous at x = 0
• D. None of these

Answer 1

The correct option is C f(x) is continuous at x = 0

$f\left(0\right)=0;f(0-)={lim}_{x\to 0}\frac{-h}{e^{\frac{-1}{k}}+1}={lim}_{k\to 0}\frac{-h}{1+\frac{1}{e^{\frac{1}{k}}}}=0$
$f(0+)={lim}_{k\to 0}\frac{h}{e^{\frac{1}{k}}+1}=0$