Question

Let $f:R\to [3,5]$ be a differentiable function such that $\underset{x\to \infty }{lim}\left(f\right(x)+f^{\prime }(x\left)\right)=3$ then $\underset{x\to \infty }{lim}f\left(x\right)$

• A. Can be obtained and is equal to $4$
• B. Can be obtained and is equal to $3$
• C. Can be obtained and is equal to $5$
• D. Can not be obtained from the given information

Answer 1

The correct option is B Can be obtained and is equal to $3$

$\underset{x\to \infty }{lim}\left(f\right(x)+f^{\prime }(x\left)\right)=3$
Let the limit to f(x) exist
then $\underset{x\to \infty }{lim}\left(f\right(x\left)\right)+\underset{x\to \infty }{lim}\left(f^{\prime }\right(x\left)\right)=3$
Let ${lim}_{x\to \infty }\left(f^{\prime }\right(x\left)\right)=k$
then${lim}_{x\to \infty }\left(f\right(x\left)\right)=kx+c$, on integrating
But if the limit of f(x) is a constant, then $k=0$
${lim}_{x\to \infty }\left(f^{\prime }\right(x\left)\right)=0$
Hence
${lim}_{x\to \infty }\left(f\right(x\left)\right)=3$