Question

If $f\text{'}\left(x\right)=f\left(x\right)$,$f\left(0\right)=1$, then$\frac{\underset{x\to 0}{\lim }[f(x)-1]}{x}=$

• A. $0$
• B. $1$
• C. $-1$
• D. $2$

Answer 1

The correct option is A

$0$

Explanation for the correct option:

Find the value of $\frac{\underset{x\to 0}{\lim }[f(x)-1]}{x}$:

$$\begin{gathered} f\text{'}\left(x\right)=f\left(x\right) \\ f\left(0\right)=1 \end{gathered}$$

Applying L-Hospitals rule in $\frac{\underset{x\to 0}{\lim }[f(x)-1]}{x}$ we get,

$$\begin{gathered} \underset{x\to 0}{\lim }\frac{f\text{'}\left(x\right)-0}{1}=f\text{'}\left(0\right) \\ \Rightarrow f\text{'}\left(0\right)=1\left[\mathrm{Given}\right] \end{gathered}$$

Hence, the correct option is A.