One-sided limits and existence of the limit for h(x) = |x|/x at x = 0

For $x < 0$, we have $|x| = -x$, so $h(x) = \dfrac{-x}{x} = -1$ for every negative $x$. As $x \to 0^-$, the values of $h(x)$ stay fixed at $-1$, so $\lim_{x\to 0^-} h(x) = -1$.

For $x > 0$, we have $|x| = x$, so $h(x) = \dfrac{x}{x} = 1$ for every positive $x$. As $x \to 0^+$, the values of $h(x)$ stay fixed at $1$, so $\lim_{x\to 0^+} h(x) = 1$. Since the left-hand limit ($-1$) and the right-hand limit ($1$) are not equal, the two-sided limit $\lim_{x\to 0} h(x)$ does not exist.

The two-sided limit fails to exist because the left-hand and right-hand limits are unequal. The left-hand limit is negative one and the right-hand limit is positive one, so they do not match. This is not a case of unbounded growth or oscillation, since the function settles on a fixed value from each side. Because the one-sided limits disagree, the overall limit at zero does not exist.