Question

The graph of function $y=f\left(x\right)$ has unique tangent at the point $(a,0)$ through which the graph passes.

Then, evaluate $\underset{x\to a}{lim}\frac{{\log }_e\{1+6f(x\left)\right\}}{3f\left(x\right)}$.

Answer 1

Given $f\left(x\right)$ has unique tangent at $(a,0)$
$\therefore f\left(a\right)=0$ and $f^{\prime }\left(x\right)$ exists
Consider $\underset{x\to a}{lim}\frac{{\log }_e\{1+6f(x\left)\right\}}{3f\left(x\right)}$ $\left(\frac{0}{0}form\right)$
$=\underset{x\to a}{lim}\frac{\left(\frac{1}{1+6f\left(x\right)}\right)6f^{\prime }\left(x\right)}{3f^{\prime }\left(x\right)}$ (using L'Hospital's rule)
$=\underset{x\to a}{lim}\frac{2}{\{1+6f(x\left)\right\}}$
$=\frac{2}{1+6f\left(a\right)}$
$=\frac{2}{1+0}=2$ $[\because f(a)=0]$