Question

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

Answer 1

By L - hospital rule

$\underset{x\to a}{Lt}\frac{\text{log}(1+6f(x\left)\right)}{3f\left(x\right)}=\underset{x\to a}{Lt}\frac{\frac{6f^{\prime }\left(x\right)}{1+6f\left(x\right)}}{3f^{\prime }\left(x\right)}\text{given}f\left(a\right)=0$

$=\frac{6f^{\prime }\left(x\right)}{3f^{\prime }\left(x\right)}=2$