Show that $x+x^2\sin\frac{1}{x}$ is not increasing near $x=0$. [duplicate]

This problem is from the question given in my classes: we are asked to show that $h(x)=\frac{x}{2}+x^2\sin(\frac{1}{x})$ is not increasing near $0$, which can be proved by the fact that $h$ is $C^1$ on $(0,\infty)$ and $h'(\frac{1}{2n\pi})=-\frac{1}{2}$. By applying this technique, we know that $cx+x^2\sin \frac{1}{x}$ is not increasing near $0$ for any $c<1$ and is increasing near $0$ for any $c>1$.

However, the case $c=1$ somewhat subtle, it can be shown that $1+2x\sin \frac{1}{x}-\cos\frac{1}{x}$ (the derivative of $x+x^2\sin \frac{1}{x}$) has limit inferior $0$ as $x\to 0$ since $$\lim_{n\to \infty} 2x\sin\frac{1}{x}= 0,$$ albeit this does not give us any useful information (at least in my point of view). Therefore, I look forward to any solution to this problem.