Limits

One-Sided Limits

Evaluate a limit approached from one side only, where the function behaves differently from each direction.

Example

Compare side behavior and state whether the two-sided limit exists.

highlighted = computed this step

Step 1 — Set up

Set up the right-hand limit at x = 0.

lim⁡x→0+e−1x\lim_{x\to 0 ^{+}} e^{-\frac{1}{x}}

Step 2 — Right-hand limit

From the right, the limit is 0.

x→0+⇒0x\to 0 ^{+}\Rightarrow \hlmath{0}

Step 3 — Left-hand check

From the left, the values grow without bound.

x→0−⇒∞x\to \hl{0} ^{-}\Rightarrow\infty

Step 4 — Existence verdict

The one-sided limits differ, so the two-sided limit does not exist.

two-sided DNE\hl{two-sided DNE}
one-sided-limit A one-sided limit lim_{x→a⁺} f(x) considers only x values approaching a from the right. Analyze the exponent or argument behavior as x approaches a from that side.