Mathematics - Limits Question with Solution | TestHub

Clearly, $\alpha$ is a repeated root of $f(x)=0$.

$$\underset{x\to \alpha }{\lim }\frac{f(x)}{f^{\prime }(x)}\left(\frac{f^{\prime }(x)}{f(x)}-\left\{\frac{f^{\prime }(x)}{f(x)}\right\}\right)$$

$$\Rightarrow \underset{x\to \alpha }{\lim }\left(1-\frac{f(x)}{f^{\prime }(x)}\left\{\frac{f^{\prime }(x)}{f(x)}\right\}\right)$$

Since ${\lim }_{x\to \alpha }\frac{f(x)}{f^{\prime }(x)}=0$ and $\left\{\frac{f^{\prime }(x)}{f(x)}\right\}$ is a bounded function, the expression simplifies.