AP Calculus AB and BC

One-Sided vs Two-Sided Limit

Q: Does a limit exist if the left and right limits are different?

No. The two-sided limit exists only when both one-sided limits exist and match. Different values mean the limit does not exist, and you report the two one-sided values as the justification.

Q: Can a limit exist at the endpoint of a domain?

The one-sided limit can, and that is the right question to ask there. Asking for a two-sided limit at x = 0 for \sqrt{x} is meaningless, because the definition needs values on both sides. Continuity at an endpoint is stated with the one-sided limit for exactly this reason.

Q: When should I check one-sided limits?

Any time the function changes character at the point: a piecewise definition splitting there, an absolute value, a greatest integer function, a vertical asymptote, or a square root whose domain stops. Elsewhere direct substitution settles both sides at once.

A two-sided limit exists exactly when both one-sided limits exist and are equal, and its value is their common value. If the two sides disagree the two-sided limit does not exist, even though both one-sided limits do. At the endpoint of a domain only one side is available, and that is normal.

One-sided limit

Use when: The function is approached from a single direction, because the domain stops there, or because a piecewise rule or an absolute value changes the behaviour at that point.

Two-sided limit

Use when: The point sits inside the domain with function values on both sides, and you need the single value that continuity, differentiability, and the limit laws all refer to.

Side by side

	One-sided limit	Two-sided limit
Notation	$\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$	$\lim_{x \to a} f(x)$
Directions of approach	One	Both, and the two must agree
At $x = 0$ for $f(x) = \frac{\lvert x \rvert}{x}$	$-1$ from the left, $1$ from the right	Does not exist, the sides disagree
At $x = 0$ for $f(x) = \sqrt{x}$	Exists and equals $0$ from the right	Not available, there is nothing on the left
Role in continuity	Defines continuity at the endpoint of a closed interval	Must equal $f(a)$ for continuity at an interior point

The entire relationship is one biconditional, and it runs in both directions. If the two-sided limit is $L$ , then approaching from either side alone also gives $L$ . If both one-sided limits exist and share a value, the two-sided limit exists and takes that value. Nothing else is needed, and nothing weaker will do.

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

Domain endpoints are where students expect trouble and there is none. The function $f(x) = \sqrt{x}$ has no values to the left of $0$ , so $\lim_{x \to 0^+} \sqrt{x} = 0$ is the whole story, and continuity on $[0,\infty)$ is defined using that one-sided limit. Piecewise rules are the other common setting: if $f(x) = x+1$ for $x < 2$ and $f(x) = x^2$ for $x \ge 2$ , the left limit at $2$ is $3$ and the right limit is $4$ , so the two-sided limit fails and the graph jumps.

The mistake: answering a two-sided question from one side

Consider $\lim_{x \to 0} \frac{1}{x}$ . The right-hand values increase without bound while the left-hand values decrease without bound, so the two sides never settle on a common value and the two-sided limit does not exist, not even as an infinite limit. Checking one side, seeing a clean behaviour, and reporting it as the answer is the most common way this goes wrong, and the same trap sits at any vertical asymptote where the function changes sign across the point, and at every jump. Compare $\frac{1}{x^2}$ at $0$ , where both sides run to $+\infty$ : there the sides agree, so a one-sided glance happens to land on the right answer.

Frequently asked questions

Does a limit exist if the left and right limits are different?

No. The two-sided limit exists only when both one-sided limits exist and match. Different values mean the limit does not exist, and you report the two one-sided values as the justification.

Can a limit exist at the endpoint of a domain?

The one-sided limit can, and that is the right question to ask there. Asking for a two-sided limit at $x = 0$ for $\sqrt{x}$ is meaningless, because the definition needs values on both sides. Continuity at an endpoint is stated with the one-sided limit for exactly this reason.

When should I check one-sided limits?

Any time the function changes character at the point: a piecewise definition splitting there, an absolute value, a greatest integer function, a vertical asymptote, or a square root whose domain stops. Elsewhere direct substitution settles both sides at once.

In the CED: Unit 1: Limits and Continuity