Question

The graph of function $y=f\left(x\right)$ has a unique tangent at $(e^a,o)$ through which the graphs passes, then $\underset{x\to e^a}{lim}\frac{\log (1+7f(x\left)\right)-\sin \left(f\right(x\left)\right)}{3f\left(x\right)}$

• A. $1$
• B. $2$
• C. $7$
• D. None of these

Answer 1

The correct option is B $2$

As $x\to e^a,y\to 0$

$f\left(x\right)\to 0$ (point given as $(e^a,0)$)

$\underset{x\to e^a}{lim}\frac{\log (1+7f\left(x\right))-\sin \left(f\left(x\right)\right)}{3f\left(x\right)}$

$=\underset{f\left(x\right)\to 0}{lim}\frac{\left(7f\left(x\right)\right)-\frac{f\left(x\right)}{1!}}{3f\left(x\right)}$

$=\log (1+P)=P-\frac{P^2}{2}+\frac{P^3}{3}+....$

$=\sin P=\frac{P}{1!}-\frac{P^3}{3!}+\frac{P^5}{5!}-....$

$\underset{f\left(x\right)\to 0}{lim}\frac{7-1}{3}=\frac{6}{3}=2$ (Higher powers of $f\left(x\right)$ will be equal to $0$)