Limits & Continuity

Related Tools

Related Labs

Related Worksheets

Related Cheat Sheets

Limits describe what a function is doing as the input gets close to a certain value, even if the function is not defined exactly there. They are one of the main ideas that make calculus possible because they let us study change, motion, and behavior near tricky points. Continuity builds on limits by asking whether a function has any breaks, jumps, or holes. Infinite behavior adds another important case by showing what happens when function values grow without bound near certain inputs.

To analyze a limit, students often compare the left-hand and right-hand behavior of a graph or expression as $x$ approaches a value. A function is continuous at $x = a$ when $f(a)$ exists, $\lim_{x\to a} f(x)$ exists, and both are equal. If one-sided limits disagree, the limit does not exist, and if values shoot upward or downward without bound, the function may have a vertical asymptote. These ideas connect graphs, tables, and algebraic rules into one framework for understanding functions.

Key Facts

$\lim_{x\to a} f(x) = L$ means $f(x)$ gets arbitrarily close to $L$ as $x$ gets close to $a$
$\lim_{x\to a^-} f(x)$ and $\lim_{x\to a^+} f(x)$ must be equal for $\lim_{x\to a} f(x)$ to exist
A function is continuous at $x = a$ if $f(a)$ exists, $\lim_{x\to a} f(x)$ exists, and $\lim_{x\to a} f(x) = f(a)$
If $\lim_{x\to a} f(x) = \infty$ or $\lim_{x\to a} f(x) = -\infty$ , then $f(x)$ has infinite behavior near $x = a$
A common vertical asymptote occurs when $f(x) = \frac{1}{x - a}$ , where $\lim_{x\to a^-} f(x) = -\infty$ and $\lim_{x\to a^+} f(x) = \infty$
For a removable discontinuity, simplifying first can reveal the limit, such as $\frac{x^2 - 1}{x - 1} = x + 1$ for $x \neq 1$ , so $\lim_{x\to 1} \frac{x^2 - 1}{x - 1} = 2$

Vocabulary

Limit: The value a function approaches as the input gets close to a given number.
One-sided limit: A limit found by approaching a point only from the left or only from the right.
Continuous: A function is continuous at a point if there is no break there and the limit equals the function value.
Discontinuity: A point where a function has a break, hole, jump, or other failure of continuity.
Vertical asymptote: A vertical line $x = a$ that a graph approaches when function values increase or decrease without bound.

Common Mistakes to Avoid

Assuming the function value and the limit must always be the same, which is wrong because a limit can exist even when $f(a)$ is missing or different.
Checking only one side of a point, which is wrong because a two-sided limit exists only if the left-hand and right-hand limits match.
Canceling terms at a discontinuity without noting the restriction, which is wrong because simplification can hide a hole where the original function was undefined.
Saying a limit does not exist whenever the graph goes to infinity, which is wrong because this usually means the function has infinite behavior rather than approaching a finite number.

Practice Questions

1 Find $\lim_{x\to 3} (2x + 5)$ .
2 Evaluate $\lim_{x\to 2} \frac{x^2 - 4}{x - 2}$ .
3 A graph has $\lim_{x\to 4^-} f(x) = 7$ and $\lim_{x\to 4^+} f(x) = 2$ . Explain whether $\lim_{x\to 4} f(x)$ exists and what this says about continuity at $x = 4$ .

Sign in to save