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

Given $f(x) = \sin[x]$ and $g(x) = |x|$.

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

For LHL, as $x \to 0^{-}$, $[x] = -1$ and $|x| = -x$.

$\lim_{x \to 0^{-}} f(x) g(x) = \lim_{x \to 0^{-}} (\sin(-1))(-x) = (-\sin 1)(0) = 0$

For RHL, as $x \to 0^{+}$, $[x] = 0$ and $|x| = x$.

$\lim_{x \to 0^{+}} f(x) g(x) = \lim_{x \to 0^{+}} (\sin 0)(x) = (0)(0) = 0$

Since LHL = RHL = $0$, the limit exists and is equal to $0$.

Answer: $0$