Limits in calculus, explained with examples

By Ashraf Bhura, Co-Founder, STEM-Tutor · Updated October 2026

Limits are the foundation of calculus. Derivatives and integrals are both defined using limits, so understanding them makes the rest of the course make sense.

What a limit means

The limit of f(x) as x approaches a, written lim_x→a f(x) = L, means that f(x) gets as close as we like to L when x is close enough to a, but not equal to a. The key point is that the limit describes what happens near a, not at a. A function can have a limit at a point where it is undefined, and its value at a point can differ from its limit there.

Imagine walking towards a bridge along a road. The limit is where the road is heading, even if the bridge itself is missing. That is exactly the situation in a derivative, where we want the gradient of a chord as its length shrinks towards zero, even though a chord of length zero makes no sense.

One-sided limits

The left-hand limit, lim_x→a⁻ f(x), looks at x values just below a. The right-hand limit, lim_x→a⁺ f(x), looks at values just above a. The two-sided limit exists only when both one-sided limits exist and are equal. Piecewise functions, absolute values and jumps on graphs are the usual places where they differ.

A method for evaluating limits

1. Substitute. If the function is continuous at a, the limit is just f(a).
2. If you get 0/0, the expression is indeterminate. Simplify by factoring, rationalising or using a known limit, then substitute again.
3. If you get a non-zero number over 0, the function heads to positive or negative infinity. Check each side to see which.
4. For limits at infinity, divide by the highest power of x in the denominator.
5. If nothing else works, try L'Hôpital's rule or the squeeze theorem.

Worked examples

Example 1: direct substitution

Find lim_x→2 (x² + 3x).

• Polynomials are continuous, so substitute: 4 + 6 = 10.

Example 2: factoring

Find lim_x→3 (x² − 9)/(x − 3).

• Substitution gives 0/0. Factor: (x − 3)(x + 3)/(x − 3) = x + 3 for x ≠ 3.
• Limit = 3 + 3 = 6. The graph is the line y = x + 6 with a hole at x = 3.

Example 3: rationalising

Find lim_x→0 (√(x + 4) − 2)/x.

• Multiply top and bottom by √(x + 4) + 2: the top becomes (x + 4) − 4 = x.
• So the expression is 1/(√(x + 4) + 2), and the limit is 1/(2 + 2) = 1/4.

Example 4: one-sided limits

f(x) = x + 1 for x < 2 and f(x) = 5 − x for x ≥ 2. Does lim_x→2 f(x) exist?

• Left limit: 2 + 1 = 3. Right limit: 5 − 2 = 3.
• Both equal 3, so the limit exists and equals 3.

Example 5: a vertical asymptote

Find lim_x→1⁺ 1/(x − 1) and lim_x→1⁻ 1/(x − 1).

• From the right, x − 1 is a tiny positive number, so the expression goes to +∞.
• From the left, it is a tiny negative number, so it goes to −∞. The two-sided limit does not exist.

Example 6: a limit at infinity

Find lim_x→∞ (3x² + 1)/(2x² − 5x).

• Divide top and bottom by x²: (3 + 1/x²)/(2 − 5/x).
• As x → ∞, the small terms vanish: the limit is 3/2, a horizontal asymptote at y = 1.5.

Example 7: a special trig limit

Find lim_x→0 sin(5x)/x.

• Use lim_θ→0 sinθ/θ = 1 (with θ in radians). Write sin(5x)/x = 5 × sin(5x)/(5x).
• Limit = 5 × 1 = 5.

Example 8: the squeeze theorem

Find lim_x→0 x² sin(1/x).

• Since −1 ≤ sin(1/x) ≤ 1, we have −x² ≤ x² sin(1/x) ≤ x².
• Both outer functions tend to 0, so the limit is squeezed to 0.

Reading limits from a graph

Many AP questions show a graph and ask for limits. Trace the curve towards the x value from the left with one finger and from the right with another. If both fingers arrive at the same height, that height is the limit, even if there is an open circle there or a filled dot somewhere else. A filled dot shows the function value; an open circle shows a point that is missing. Keep these separate and graph questions become quick marks.

Limits and continuity

A function f is continuous at x = a when three things hold: f(a) is defined, lim_x→a f(x) exists, and the two are equal. AP free-response questions often ask you to justify continuity using exactly these three conditions, so write all three. There are three kinds of discontinuity to recognise: a removable discontinuity (a hole, where the limit exists but does not match the value), a jump (left and right limits differ), and an infinite discontinuity (a vertical asymptote).

Limits and the derivative

The derivative is defined as a limit: f'(x) = lim_h→0 (f(x + h) − f(x))/h. AP questions sometimes give you a limit in this form and ask what it represents. For example, lim_h→0 ((2 + h)³ − 8)/h is the derivative of x³ at x = 2, which is 3 × 2² = 12. Recognising this saves a long calculation.

Common mistakes

• Saying a limit does not exist just because substitution gives 0/0; 0/0 means more work is needed.
• Using degrees in trig limits; sin x/x → 1 only in radians.
• Forgetting to check both sides at a jump or an asymptote.
• Confusing the value f(a) with the limit as x approaches a.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Are limits on the AP Calculus exam?

Yes. Limits and continuity are Unit 1 of both AB and BC, worth roughly 10 to 12% of the AB exam, and limit ideas appear throughout the rest of the course.

Can I use L'Hôpital's rule for every 0/0 limit?

Only when its conditions hold, and on free-response questions you must state that the limit is an indeterminate form first. Simple algebra is often quicker.

Key takeaways

• A limit describes behaviour near a point, not at it.
• Try substitution first; 0/0 means simplify.
• Two-sided limits need matching one-sided limits.
• For limits at infinity, compare the highest powers.
• Continuity needs a value, a limit, and the two to be equal.

Related guides

• L'Hôpital's rule
• AP Calculus AB units
• AP Calculus derivative rules