Question

Let $f$ be a continuous function satisfying $f(x+y)=f\left(x\right)+f\left(y\right)$ for all $x,y\in R$ and $f\left(1\right)=5$ then $\underset{x\to 4}{lim}f\left(x\right)$ is equal to

• A. $4$
• B. $80$
• C. $0$
• D. none of these

Answer 1

The correct option is D none of these

$f(x+y)=f\left(x\right)+f\left(y\right)$
Substituting $x=y=1$
$f\left(2\right)=f\left(1\right)+f\left(1\right)=2f\left(1\right)=2\times 5=10$
Now substituting $x=y=2$
$f\left(4\right)=f\left(2\right)+f\left(2\right)=2f\left(2\right)=2\times 10=20$
$\Rightarrow \underset{x\to 4}{lim}f\left(x\right)=f\left(4\right)=20$
Hence, option 'D' is correct.