Math Problem Solver

The two-sided limit $\lim_{x \to 0}{\frac{\sin\left( x \right) - \cos\left( x \right)}{4 x}}$ does not exist because the directional limits from the right and the left result in different values.

Right limit:
$\displaystyle\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} - \cos{\left(x \right)}}{4 x}\right) = -\infty$

Left limit:
$\displaystyle\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} - \cos{\left(x \right)}}{4 x}\right) = \infty$