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Calculus fundamentals

Limits and Continuity

Learn how limits describe nearby behavior, how to test continuity, and how to handle common forms like piecewise and removable discontinuities.

How this page is maintained

Written for learners, checked against the sources below, and reviewed every year. Last reviewed July 22, 2026.

Short answer

A limit is the value a function approaches as x gets close to a target input. A function is continuous at a point when the two-sided limit exists, equals the function value, and both come from the same input point.

  • A limit concerns nearby values, not only the value at the point.
  • Continuity at x = a requires f(a) to exist and match the limit as x approaches a.
  • Factoring and rationalizing can remove algebraic blocks before taking a limit.

Read limits as behavior near an input

When you write lim x->a f(x), you are asking what y-values f(x) gets close to as x gets close to a from either side. The function might be undefined at x = a and still have a limit there.

One-sided limits help diagnose breaks. If left and right limits are different, the two-sided limit does not exist.

Use continuity tests and basic algebra cleanup

For polynomials and many common functions, substitution works directly. For rational expressions, a zero denominator can hide a common factor that cancels.

In piecewise functions, check left and right behavior at the boundary point, then compare with the declared value at that point.

  • Check left limit and right limit first.
  • Then compare to the actual function value.
  • State clearly whether the issue is removable, jump, or infinite behavior.

Evaluate a removable discontinuity limit

Find lim x->3 (x^2 - 9)/(x - 3).

  1. Factor the numerator: x^2 - 9 = (x - 3)(x + 3).
  2. Cancel (x - 3) for x != 3, giving x + 3.
  3. Take the limit of x + 3 as x->3.
Result: The limit is 6 even though the original expression is undefined at x = 3.

Common mistakes

  • Plugging in first and stopping at 0/0 without simplifying.
  • Assuming continuity just because a graph looks smooth.
  • Ignoring one-sided limits at piecewise boundaries.
  • Confusing f(a) with lim x->a f(x).

Try one

If lim x->2 f(x) = 5 and f(2) = 8, is f continuous at x = 2?

No. Continuity requires f(2) to equal the limit, and 8 does not equal 5.

Sources

Core derivative conceptDerivative definitionDifferentiation toolkitDerivative rulesComposite functionsChain rule

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