Question

Let $f(x)=\begin{cases}x \sin \left(\frac{1}{x}\right)+\sin \left(\frac{1}{x^{2}}\right), & x \neq 0 \\ 0, & x=0\end{cases}$
then $\displaystyle\lim _{x \rightarrow \infty} f(x)$ equals

• A. 0
• B. -1 / 2
• C. 1
• D. None of these

Answer 1

Answer: (C)

$\displaystyle\lim _{x \rightarrow \infty} f(x)=\displaystyle\lim _{x \rightarrow \infty} x \sin \left(\frac{1}{x}\right)+\sin \left(\frac{1}{x^{2}}\right)$
$=\displaystyle\lim _{x \rightarrow \infty} \frac{\sin \left(\frac{1}{x}\right)}{\left(\frac{1}{x}\right)}+\displaystyle\lim _{x \rightarrow \infty} \sin \left(\frac{1}{x^{2}}\right)$
$=1+0=1$