Limits
A limit describes the value that a function approaches as the input variable gets arbitrarily close to a specific point. In mathematical notation, lim(x→a) f(x) = L means that f(x) approaches the value L as x approaches a. Limits form the foundation of calculus and appear throughout A-level Further Mathematics and university-level mathematical analysis.
Why it matters
Limits provide the mathematical framework for understanding rates of change and areas under curves, making them essential for physics, engineering, and economics. In Year 13 Further Mathematics, students encounter limits when studying differentiation and integration. Engineers use limits to model the behaviour of systems approaching critical thresholds — for instance, calculating the maximum load a bridge can handle before failure, or determining the efficiency of an engine as fuel mixture ratios approach optimal values. Financial analysts apply limits to understand compound interest growth over infinite time periods, whilst physicists rely on limits to describe instantaneous velocity and acceleration. The concept appears in population dynamics models, where limits help predict carrying capacity in ecological systems. Modern computer graphics and animation software uses limits extensively for smooth curve generation and realistic motion simulation.
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.
Worked examples
Find lim(x→4) of (-2x − 5)
Answer: -13
- Use direct substitution (innsetting): replace x with the value → f(4) = -2·4 − 5 — Since f(x) = -2x − 5 is a polynomial, we can substitute x = 4 directly.
- Calculate the result → lim(x→4) = -13 — -2 × 4 = -8, then -8 − 5 = -13.
Find lim(x→2) of (x² − 4)/(x − 2)
Answer: 4
- Try direct substitution → (2² − 4)/(2 − 2) = 00 — We get the indeterminate form 0/0, so we need to simplify.
- 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.
- Cancel the common factor (forkorte) → (x − 2)(x + 2) / (x − 2) = x + 2 — After cancelling (x − 2), we have f(x) = x + 2.
- Now substitute x = 2 → lim(x→2) = 2 + 2 = 4 — The limit is 4.
Find lim(x→∞) of (2 x - 2) / (x2 + 1)
Answer: 0
- Identify the degrees of numerator and denominator → Numerator: 2 x - 2, Denominator: x2 + 1 — For limits at infinity, compare the leading terms of the polynomials.
- 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.
- State the limit → lim(x→∞) = 0 — The limit is 0.
Common mistakes
- Attempting to evaluate lim(x→2) (x²−4)/(x−2) by direct substitution gives 0/0 = 0, when the correct limit is 4 after factoring and cancelling.
- Incorrectly claiming that lim(x→∞) (3x²+1)/(2x²−5) equals ∞ when it actually equals 3/2 by comparing leading coefficients.
- Writing lim(x→0) sin(x)/x = 0/0 = 0 instead of recognising this standard limit equals 1.