Concept explainerCalculus IFoundational8 min

One-sided limits and jump discontinuities

A two-sided limit exists only when the function approaches the same value from the left and from the right.

LaTeX article Updated July 13, 2026

\lim_{x\to a}f(x)=L\iff\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L

Direction belongs to the input

The notation x → a⁻ means x approaches a through values smaller than a. The plus sign means values larger than a.

A one-sided limit reads the behavior on only one branch of a piecewise graph, so it can exist even when the full two-sided limit does not.

\lim_{x\to a^-}f(x)=L_-\qquad\lim_{x\to a^+}f(x)=L_+

A two-sided limit is a single claimed destination. If the left and right approaches disagree, no single number describes the nearby behavior.

Do not average the two values. The limit is about agreement, not finding a compromise between branches.

L_-=2,\ L_+=5\quad\Rightarrow\quad\lim_{x\to a}f(x)\text{ DNE}

The point f(a) may equal one side, the other side, or neither. Changing a single plotted point does not change the nearby approach.

Continuity requires all three pieces: the function is defined, the two-sided limit exists, and that limit equals the function value.

f(a)\text{ exists},\quad\lim_{x\to a}f(x)\text{ exists},\quad\lim_{x\to a}f(x)=f(a)