Let $a - 1 < 0$ and $b > 0$.
$$ f(x) = \left\{ \begin{array}{ll} x^{a-1}(\sin(x))^b\cos(1/x) & \quad x ≠ 0 \\ 0 & \quad x = 0 \end{array} \right. $$
I suspect that $\underset{x\to 0}{\lim}f(x)$ doesn't exist. But I don't know how to show it and l'hopital's rule cannot be used.