The graph of y = f(x) has unique tangent at the point (a,0) through which the graph passes. Then $\underset{x\to a}{lim} \frac{\log [1+6f(x\left)\right]}{3f\left(x\right)}=$

$=\underset{x\to a}{lim} \frac{\log [1+6f(x\left)\right]}{3f\left(x\right)}$

$$\begin{gathered} \text{ (using L'Hospital's rule) } \\ =\underset{x\to a}{lim} \frac{\left(\frac{1}{1+6f\left(x\right)}\right)6f^{\mathrm{\prime }}\left(x\right)}{3f^{\mathrm{\prime }}\left(x\right)} \end{gathered}$$

$=\underset{x\to a}{lim} \frac{2}{\{1+6f(x\left)\right\}}$

$=\frac{2}{1+6f\left(a\right)}$

$=\frac{2}{1+0}=2 \left[\mathrm{f}\right(\mathrm{a})=0]$