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Question: Answered & Verified by Expert
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?
MathematicsLimitsKVPYKVPY 2013 (SB/SX)
Options:
  • A $\lim _{x \rightarrow \infty} x \sin (1 / x) f(x)=M$
  • B $\lim _{x \rightarrow \infty} \sin (f(x))=\sin M$
  • C $\lim _{x \rightarrow \infty} x \sin \left(e^{-x}\right) f(x)=M$
  • D $\lim _{x \rightarrow \infty} \frac{\sin x}{x} \cdot f(x)=0$
Solution:
1349 Upvotes Verified Answer
The correct answer is: $\lim _{x \rightarrow \infty} x \sin \left(e^{-x}\right) f(x)=M$
$\lim _{x \rightarrow \infty} x \sin \left(e^{-x}\right) f(x)=\lim _{x \rightarrow \infty} \frac{\sin \left(e^{-x}\right)}{e^{-x}} \frac{x}{e^{x}} f(x)$
$=1 \times(0) \times M=0$

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