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(C) $\lim _{x \rightarrow \infty} x \sin \left(\frac{1}{x}\right) f(x) = \lim _{x \rightarrow \infty} \frac{\sin (1/x)}{1/x} \cdot f(x) = 1 \cdot M =

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

Vedclass · Answer

(C) $\lim _{x \rightarrow \infty} x \sin \left(\frac{1}{x}\right) f(x) = \lim _{x \rightarrow \infty} \frac{\sin (1/x)}{1/x} \cdot f(x) = 1 \cdot M = M$. This is true.$(B)$ Since $\sin(x)$ is a continuous function,$\lim _{x \rightarrow \infty} \sin(f(x)) = \sin(\lim _{x \rightarrow \infty} f(x)) = \sin M$. This is true.$(C)$ $\lim _{x \rightarrow \infty} x \sin \left(e^{-x}\right) f(x) = \lim _{x \rightarrow \infty} (x e^{-x}) \cdot \frac{\sin (e^{-x})}{e^{-x}} \cdot f(x)$. Since $\lim _{x \rightarrow \infty} x e^{-x} = 0$,the limit is $0 \cdot 1 \cdot M = 0$. Thus,the statement that the limit is $M$ is false.$(D)$ $\lim _{x \rightarrow \infty} \frac{\sin x}{x} f(x) = (\lim _{x \rightarrow \infty} \frac{\sin x}{x}) \cdot (\lim _{x \rightarrow \infty} f(x)) = 0 \cdot M = 0$. This is true.

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

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