Calculus · step 1 of 13

Limits & continuity

The limit of f as x → a describes what f approaches near a, deliberately ignoring the value at a itself. Continuity is the statement that these agree: f is continuous at a when the limit exists, f(a) exists, and they are equal. Most exam limits are 0/0 indeterminate forms, cleared by factoring, rationalising, or recognising a standard limit like sin x / x → 1.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Substitute firstIf direct substitution gives a number, that is the limit. Only an indeterminate form requires work.
Clear 0/0 by factoringCancel the common factor causing the zero. For roots, multiply by the conjugate to rationalise.
Check one-sided limits when neededFor piecewise functions and absolute values the two sides may differ, in which case the limit does not exist.
Test continuity with the three conditionsf(a) defined, limit exists, and the two are equal. Name which condition fails.

Worked example

Evaluate lim_{x→3} (x² − 9)/(x − 3).

Substitution gives 0/0 — indeterminate, so more work is needed.
Factor the numerator: x² − 9 = (x−3)(x+3).
For x ≠ 3 the expression equals x + 3, and the limit ignores x = 3 itself.
So the limit is lim_{x→3} (x + 3).

Answer. 6. The function is undefined at x = 3, but the limit exists — a removable discontinuity.

Where marks get dropped

These are the specific errors that cost credit on limits & continuity questions — QED's rubric penalises each of them separately.

Concluding the limit does not exist because substitution gives 0/0. That form is indeterminate, not undefined — it means you must simplify.
Cancelling and then evaluating at the removed point without noting the function is undefined there. The limit exists; the value does not.
Ignoring one-sided behaviour for |x| or piecewise functions, where the left and right limits genuinely differ.

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Limits & continuity — frequently asked questions

What is an indeterminate form?

An expression like 0/0, ∞/∞, 0·∞ or 1^∞ whose value is not determined by the form alone. Different functions with the same form give different limits.

What is a removable discontinuity?

A point where the limit exists but the function is undefined or has the wrong value. Redefining f at that single point makes it continuous.

Do I need the ε–δ definition?

For computation, no — the limit laws suffice. For proofs that a specific limit equals a value, some courses require it, so check what your paper asks.

The rest of Calculus

Limits, derivatives, integrals and their applications. Each subtopic below has its own method, worked example and mark-losing traps.

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