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Limits and Continuity

Standard limits, series expansions and the one-to-the-power-infinity rule; continuity is one equation. Counting points of discontinuity takes the longest.

Questions in the bank
136
q/paper in 2025–26
0.92
Numeric answer
25%
Notes pages
8

Tier: Long tail

When you’ll see it

A limit that starts as 0/0, ∞/∞, ∞ − ∞ or 1^∞, or a piecewise function whose constants are fixed by continuity.

How this chapter is tested

Most limits reduce to a short list: sin x/x → 1, (eˣ − 1)/x → 1, ln(1 + x)/x → 1, and the 1^∞ rule — if f → 1 and g → ∞, f^g tends to e raised to lim g(f − 1). Series expansions of sin x, cos x, eˣ and ln(1 + x) settle the rest, and they are the quickest route when a question asks for the constants that make a limit finite.

Limits at infinity compare the highest powers or rationalise a surd. A limit of a sum either closes the sum first or reads it as a Riemann sum, which is Definite Integration in another form. Limits via derivatives spot a difference quotient or use L'Hospital's rule, and an integral with a variable limit is differentiated by the Leibniz rule.

Continuity at a point is usually one equation: left limit = right limit = f(a), solved for the constants. Counting points of discontinuity or non-differentiability takes longer — list every jump of a greatest-integer term and every corner of a modulus, then test each point, because a zero factor can hide a jump. These counts suit numeric-answer questions.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Standard limits and algebraic forms

    sin x/x, (eˣ − 1)/x, ln(1 + x)/x; factorise or rationalise to remove the factor that gives 0.

  • Series expansions

    sin x = x − x³/6 + …, cos x = 1 − x²/2 + …, eˣ = 1 + x + x²/2 + …; set the lower coefficients to zero for a finite limit.

  • 1^∞ limits

    If f → 1 and g → ∞, then f^g → e raised to lim g(f − 1).

  • Limits at infinity and of sums

    Divide by the highest power; (1/n) Σ f(k/n) for k = 1 to n tends to ∫ f(x) dx from 0 to 1.

  • Limits via derivatives and integrals

    A difference quotient is a derivative; L'Hospital needs 0/0 or ∞/∞; the derivative of ∫ g(t) dt from a to u(x) is g(u(x)) · u′(x).

  • Greatest integer and one-sided limits

    Ask which side of the integer the inside approaches; |x| and √(x²) need both sides checked.

  • Continuity at a point

    Left limit = right limit = f(a); solve for the unknown constants.

  • Counting discontinuities

    List jumps of [ ] and corners of | |, then test each; for f(g(x)) also solve g(x) = each bad point of f.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • The argument does not match

    sin 3x/x tends to 3, not 1. Make the angle and the denominator the same before using the standard limit.

  • Expanding too little

    Stop a series too early and everything cancels to 0/0 again. Expand each function up to the power of x in the denominator.

  • Infinity minus infinity is not 0

    √(x² + x) − x tends to 1/2. Rationalise before comparing.

  • A zero factor hides a jump

    x[x] is continuous at 0 because the jump of [x] is multiplied by 0. Test each candidate point instead of counting integer crossings.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Limits and Continuity notes

Drill every Limits and Continuity question

136 questions from the bank, across 8 subtopics.

Drill the 136 questions

Drill one subtopic at a time

The 8 subtopics, in teaching order.

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