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§ Calculus

Limits

LK20.R1LK20.S13 min read

Limits form the cornerstone of A Level Further Mathematics and university calculus courses, yet many Year 12 students struggle with the transition from GCSE algebraic manipulation to understanding what happens as x approaches a value. The concept bridges the gap between discrete GCSE topics and continuous mathematical analysis.

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§ 01

Why it matters

Limits underpin essential calculus concepts that appear across STEM fields. In physics, velocity calculations require understanding instantaneous rates of change through limits—a car's speedometer reading at precisely 14:30 represents a limit as time intervals approach zero. Engineering students use limits to analyse structural stress points, whilst economists apply them to model marginal cost functions. A factory producing widgets might need to determine the limiting cost per unit as production approaches 10,000 items daily. Medical researchers studying drug concentration in blood use limits to model decay rates approaching zero over time. For A Level students progressing to Russell Group universities, mastering limits provides the foundation for differential calculus, integral calculus, and advanced mathematical modelling required in competitive degree programmes.

§ 02

How to solve limits

Limits

  • A limit describes the value a function approaches as x approaches a point.
  • Try direct substitution first: replace x with the target value.
  • If you get 00 (indeterminate), factor or simplify the expression and try again.
  • For polynomials and rational functions, direct substitution usually works after simplification.

Example: lim(x→2) (x² − 4)/(x − 2) = lim(x→2) (x+2) = 4.

§ 03

Worked examples

Beginner§ 01

Find lim(x→3) of (-2x − 3)

Answer: -9

  1. Use direct substitution (innsetting): replace x with the value → f(3) = -2·3 − 3 — Since f(x) = -2x − 3 is a polynomial, we can substitute x = 3 directly.
  2. Calculate the result → lim(x→3) = -9 — -2 × 3 = -6, then -6 − 3 = -9.
Easy§ 02

Find lim(x→2) of (x² − 4)/(x − 2)

Answer: 4

  1. Try direct substitution → (2² − 4)/(2 − 2) = 0/0 — We get the indeterminate form 0/0, so we need to simplify.
  2. Factor the numerator (telleren) using the difference of squares → x² − 4 = (x - 2) (x + 2) — x² − 4 = (x − 2)(x + 2) is a difference of squares.
  3. Cancel the common factor (forkorte) → (x − 2)(x + 2) / (x − 2) = x + 2 — After cancelling (x − 2), we have f(x) = x + 2.
  4. Now substitute x = 2 → lim(x→2) = 2 + 2 = 4 — The limit is 4.
Medium§ 03

Find lim(x→∞) of (4 x - 2) / (2 x2 + 1)

Answer: 0

  1. Identify the degrees of numerator and denominator → Numerator: 4 x - 2, Denominator: 2 x^2 + 1 — For limits at infinity, compare the leading terms of the polynomials.
  2. Compare leading terms (ledende ledd) → Numerator degree (1) < denominator degree (2) → 0 — When the denominator has a higher degree, the denominator grows faster and the fraction approaches 0.
  3. State the limit → lim(x→∞) = 0 — The limit is 0.
§ 04

Common mistakes

  • Students often substitute directly into indeterminate forms, writing lim(x→2) (x²-4)/(x-2) = 0/0 = 0 instead of factoring to get 4
  • Many pupils incorrectly assume lim(x→∞) (3x²+5)/(2x²-1) = 3/2 by comparing coefficients rather than leading terms, missing the correct answer of 3/2
  • Students frequently cancel factors incorrectly, writing (x²-9)/(x-3) = x²/x = x instead of properly factoring (x+3)(x-3)/(x-3) = x+3
  • Common error involves treating ∞/∞ as 1, calculating lim(x→∞) (5x+2)/(3x-7) = 1 instead of comparing degrees to get 5/3
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§ 05

Frequently asked questions

Why can't I just substitute x = 2 into (x²-4)/(x-2)?
Direct substitution gives 0/0, which is indeterminate—it doesn't equal zero. You must first simplify by factoring: x²-4 = (x-2)(x+2), then cancel the (x-2) terms to get x+2, which gives limit = 4 when x approaches 2.
How do I handle limits at infinity for rational functions?
Compare the degrees of numerator and denominator. If numerator degree < denominator degree, limit = 0. If degrees are equal, divide leading coefficients. If numerator degree > denominator degree, limit = ±∞ depending on signs.
What's the difference between f(2) and lim(x→2) f(x)?
f(2) is the actual function value at x = 2, whilst lim(x→2) f(x) describes what happens as x gets arbitrarily close to 2. The function might be undefined at x = 2 but still have a limit there.
When do I use rationalisation to solve limit problems?
Use rationalisation when you have square roots causing 0/0 indeterminate forms. Multiply numerator and denominator by the conjugate expression, then simplify. This technique often appears in Further Maths A Level papers.
Are limits only used in A Level Further Maths?
Whilst formal limit notation appears in Further Maths, the underlying concept connects to GCSE topics like asymptotes and rate of change. Understanding limits enhances comprehension of gradients and prepares students for university mathematics courses.
§ 06

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