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

• A

  $\lim _{x \rightarrow \infty} x \sin (1 / x) f(x)=M$
• B

  $\lim _{x \rightarrow \infty} \sin (f(x))=\sin M$
• Correct answer
  C

  $\lim _{x \rightarrow \infty} x \sin \left(e^{-x}\right) f(x)=M$
• D

  $\lim _{x \rightarrow \infty} \frac{\sin x}{x} \cdot f(x)=0$

Hence, the correct answer is option: 'C'.