Question

The graph of the function $y=f\left(x\right)$ has a unique tangent at the point $(e^a,0)$ through which the graph passes then $\underset{x\to e^a}{lim}\frac{lo{g}_e\{1+7f(x\left)\right\}-sinf\left(x\right)}{3f\left(x\right)}$ is

• A. $1$
• B. $2$
• C. $0$
• D. $-1$

Answer 1

Answer: (B)

$2$

Given $y=f\left(x\right)$ has a unique tangent at the point $(e^a,0)$.
So, as $x\to e^a$, $f\left(x\right)\to 0$
Now, $\underset{x\to e^a}{lim}\frac{lo{g}_e\{1+7f(x\left)\right\}-\sin f\left(x\right)}{3f\left(x\right)}$
$\underset{x\to e^a}{lim}\frac{lo{g}_e(1+7f(x\left)\right)}{3f\left(x\right)}-\frac{\sin f\left(x\right)}{3f\left(x\right)}$
$=\frac{7}{3}\underset{x\to e^a}{lim}\frac{lo{g}_e(1+7f(x\left)\right)}{7f\left(x\right)}-\frac{1}{3}\underset{x\to e^a}{lim}\frac{\sin f\left(x\right)}{f\left(x\right)}$
$=\frac{7-1}{3}=2$