Question

For the function $f\left(x\right)={(x+1)}^{1/x}$ to be continuous at $x=0$, $f\left(0\right)$ must be defined as:

• A. $f\left(0\right)=0$
• B. $f\left(0\right)=e$
• C. $f\left(0\right)=\frac{1}{e}$
• D. $f\left(0\right)=1$

Answer 1

The correct option is B $f\left(0\right)=e$

Given, $f\left(x\right)=(x+1{)}^{\frac{1}{x}}$

$\underset{x\to 0}{lim}f\left(x\right)=f\left(0\right)=\underset{x\to 0}{lim}{(1+x)}^{1/x}=e$