Question

If $f\left(x\right)$ is differentiable and strictly increasing function, then the value of $\underset{x\to 0}{\lim }\frac{\left[f\left(x^2\right)–f\left(x\right)\right]}{\left[f\right(x)–f(0\left)\right]}$ is equal to

• A. $-1$
• B. $0$
• C. $1$
• D. $2$

Answer 1

The correct option is A

$-1$

Explanation for correct option:

Find the value of the expression:

Since, $\underset{x\to 0}{\lim }\frac{\left[f\left(x^2\right)–f\left(x\right)\right]}{\left[f\right(x)–f(0\left)\right]}$ form $\left(\frac{0}{0}\right)$ at $x=0$, then

Apply L Hospital's rule in the finding expression,

$\begin{array}{rcl}\underset{x\to 0}{\lim }\frac{\left[f\left(x^2\right)–f\left(x\right)\right]}{\left[f\right(x)–f(0\left)\right]}& =& li{m}_{x\to 0}\frac{[f\text{'}(x^2)\times 2x–f\text{'}(x\left)\right]}{[f\text{'}(x\left)\right]}\\ & =& \frac{[0–f\text{'}(0\left)\right]}{f\text{'}\left(0\right)}\\ & =& \frac{-f\text{'}\left(0\right)}{f\text{'}\left(0\right)}\\ & =& -1\end{array}$

Hence, the correct option is A.