Limit Definition & Notation

Unit 1 · Limits & Continuity

What is a Limit?

The limit of f(x) as x approaches c describes where the function is heading — not the value at c itself. As x gets arbitrarily close to c, f(x) gets arbitrarily close to L. The two-sided limit exists only when the left-hand and right-hand limits are equal.

Limit Notation

Two-sided limit

\lim_{x \to c} f(x) = L

Left-hand limit

\lim_{x \to c^-} f(x) = L

Right-hand limit

\lim_{x \to c^+} f(x) = L

Limit Exists If and Only If

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

Limit Definition of Derivative (Preview)

Instantaneous rate of change at a

\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}

AP Tip: The limit describes where f(x) is heading, not the value f(c). The function does not even need to be defined at x = c for the limit to exist.

Type 1

One-Sided Limits

Evaluate left-hand and right-hand limits separately, then determine whether the two-sided limit exists by checking if both sides agree.

Example 1

Find the one-sided limits and determine whether the two-sided limit exists.

f(x) = \begin{cases} x^2 - 1 & x < 1 \\ 3x - 2 & x \geq 1 \end{cases}

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Type 2

Limits from Graphs

Read limits directly from a graph by tracing the curve toward the target x-value from each side. An open circle means the function is not defined there, but the limit can still exist.

Example 2

A function f is graphed. At x = 2, the curve approaches y = 4 from both sides, but there is an open circle at (2, 4) and a filled circle at (2, 1). State the limit and f(2).

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Type 3

Instantaneous Rate of Change

Use the limit definition to find the instantaneous rate of change. Expand, simplify, cancel the h, then substitute h = 0.

Example 3

Find the instantaneous velocity at t = 2 using the limit definition.

s(t) = 16t^2 \text{ (feet)}

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Algebraic Limit Evaluation →