# Is $\pm\infty$ the only case for one-sided limits to not exist?

Could one-sided limits not exist but not equal to $\pm\infty$

e.g. $\displaystyle \lim_{x->c^+}f(x)$ DNE and $\displaystyle \lim_{x->c^+}f(x)\neq\pm\infty$

• Consider $x \mapsto \cos(\frac{1}{x})$. Obviously if the right-handed and left-handed limits as $x \rightarrow 0$ existed, they would be equal, because the function is even. – Jannik Pitt Mar 31 '18 at 15:09

Consider the function $$f(x)= \sin(1/x)$$ on the interval $(0,1).$
$$\lim_{x\to {0^+}} f(x)$$ does not exist due to oscillations and the function is bounded.