Consider the following for the two (02) items that follow: Let the...

The given functions are $f(x) = \sin[x]$ and $g(x) = |x|$.

To find $\lim_{x \to 0} \dfrac{f(x)}{g(x)}$, we evaluate the left-hand limit (LHL) and right-hand limit (RHL) at $x = 0$.

For the right-hand limit ($x \to 0^+$):
When $0 \lt x \lt 1$, $[x] = 0$ and $|x| = x$.
$\lim_{x \to 0^+} \dfrac{\sin[x]}{|x|} = \lim_{x \to 0^+} \dfrac{\sin 0}{x} = \lim_{x \to 0^+} \dfrac{0}{x} = 0$

For the left-hand limit ($x \to 0^-$):
When $-1 \lt x \lt 0$, $[x] = -1$ and $|x| = -x$.
$\lim_{x \to 0^-} \dfrac{\sin[x]}{|x|} = \lim_{x \to 0^-} \dfrac{\sin(-1)}{-x} = \lim_{x \to 0^-} \dfrac{-\sin 1}{-x} = \lim_{x \to 0^-} \dfrac{\sin 1}{x} = -\infty$

Since the left-hand limit and right-hand limit are not equal, the limit does not exist.