Question

Let $f:R\to R$ be a function such that $\underset{x\to \infty }{lim}f\left(x\right)=M>0.$
Then which of the following is false ?

• A. $\underset{x\to \infty }{lim}x\sin \left(1/x\right)f\left(x\right)=M$
• B. $\underset{x\to \infty }{lim}\sin f\left(x\right)=M$
• C. $\underset{x\to \infty }{lim}x\sin \left(e^{-x}\right)f\left(x\right)=M$
• D. $\underset{x\to \infty }{lim}\frac{\sin x}{x}=0$

Answer 1

The correct option is B $\underset{x\to \infty }{lim}\sin f\left(x\right)=M$

A. $\underset{x\to \infty }{lim}x\sin \left(1/x\right)f\left(x\right)$

This can be written as $\underset{x\to \infty }{lim}\frac{\sin \left(1/x\right)}{\left(1/x\right)}\times f\left(x\right)=M$

B. $\underset{x\to \infty }{lim}\sin \left(f\right(x\left)\right)=\sin \left(M\right)$

C. $\underset{x\to \infty }{lim}x\sin \left(e^{-x}\right)f\left(x\right)=\underset{x\to \infty }{lim}\frac{\sin \left(e^{-x}\right)f\left(x\right)}{\left(1/x\right)}=M$

D. $\underset{x\to \infty }{lim}\frac{\sin x}{x}$

Since $\sin \left(\infty \right)$ will be a number between $-1$ and $1$, and the denominator tends to $\infty$, the limit would become tending to $0$