Question

$y=f\left(x\right)$ is a continuous function such that its graph passes through $(a,0)$.

Then $\underset{x\to a}{lim}\frac{{\log }_e(1+3f(x\left)\right)}{2f\left(x\right)}$ is?

• A. $1$
• B. $0$
• C. $\frac{3}{2}$
• D. $\frac{2}{3}$

Answer 1

The correct option is C $\frac{3}{2}$

Given curve $y=f\left(x\right)$ passes through (a,0)
$\Rightarrow f\left(a\right)=0$
Now, $\underset{x\to a}{lim}\frac{{\log }_e(1+3f(x\left)\right)}{2f\left(x\right)}$
It is of the form $\frac{0}{0}$, so applying L-Hospital's rule
$=\underset{x\to a}{lim}\frac{3f^{\prime }\left(x\right)}{(1+3f(x\left)\right)2f^{\prime }\left(x\right)}$
$=\underset{x\to a}{lim}\frac{3}{(1+3f(x\left)\right)2}=\frac{3}{2}$