Find the points of discontinuities of the function $x \sin\left(\frac{\pi}{x^2+x}\right)$ Need to find the points of discontinuities of the function $f(x) = x \sin\left(\dfrac{\pi}{x^2+x}\right)$
Obviously at zero. I can’t take the one-sided limit ($\lim{x \to 0+}$ and $x \to 0-$) any way, sin is periodic, and at zero has a million oscillations.
In zero discontinuities, how to prove it. And how to get similar limits.


 sorry for lang. Frrrrom Rrrussia)


Given $f(0)=0$, $f(x)$ will be continuous as the left and right limit abiut $x=0$ exist by sandwich theorem:$$ -x\le x\sin \frac{\pi}{x^2+x} \le x ~~~~(1)$$.
But by sandwich theorem (1) left lim is 1 and right limit is -1 about $x=-1$
the limit does not exist.So $f(x)$ is essentially discontinuous at $x=-1$.
