When a Limit Exists

A limit $\lim_{x \to a} f(x) = L$ exists only if the function approaches the same value from both sides:

\lim_{x \to a^-} f(x) = L \quad \text{and} \quad \lim_{x \to a^+} f(x) = L

The left-hand limit ( $x \to a^-$ ) looks at values from the left
The right-hand limit ( $x \to a^+$ ) looks at values from the right

If these two limits agree, the two-sided limit exists. If they differ, the two-sided limit does not exist (DNE).

This is common at jump discontinuities or in absolute value functions where behavior changes depending on direction.

Related To This Note

Links from this note

Links to this note