$\lim_{x \rightarrow \infty} x^a\sin{\frac{1}{x}}$
for this limit ,it was showed on the textbook that
$\lim_{x \rightarrow \infty} x^a\sin{\frac{1}{x}}=\begin{cases} 0 & a<1 \\ 1 & a=1 \\ \infty & a>1 \end{cases}$
in my opinion ,it's obviously that $\lim\limits_{x \rightarrow \infty}\sin{\frac{1}{x}}=0$ and that $\lim\limits_{x \rightarrow \infty}x^a=\infty$
I wonder what's wrong with my view