Question

Let $f$ be a continuous function satisfying $f\left(x\right)f\left(y\right)=f\left(x\right)+f\left(y\right)+f\left(xy\right)-2$ for all $x,y\in R$ and $f\left(2\right)=5$ then ${lim}_{x\to 4}f\left(x\right)$ is

• A. $5$
• B. $17$
• C. $-5$
• D. $21$

Answer 1

Answer: (B)

$17$

$f\left(x\right)f\left(y\right)=f\left(x\right)+f\left(y\right)+f\left(xy\right)-2$
Substituting $x=y=2$
$f\left(2\right)f\left(2\right)=f\left(2\right)+f\left(2\right)+f\left(4\right)-2$
$\Rightarrow 5\times 5=5+5+f\left(4\right)-2\Rightarrow f\left(4\right)=17$
$\Rightarrow \underset{x\to 4}{lim}f\left(x\right)=f\left(4\right)=17$
Hence, option 'B' is correct.