Limits

Limits describe what a function approaches as its input approaches a value. Learn the intuitive and formal ideas, one-sided and two-sided limits, infinite limits and limits at infinity, algebraic techniques (factorising, rationalising), standard limits such as sin x/x, indeterminate forms and L'Hôpital's rule.

Calculus is built on limits. Instantaneous velocity is the limit of average velocity over shorter and shorter time intervals; the area under a curve is the limit of sums of thinner rectangles. Limits also describe steady-state values and behaviour near singularities.

An analogy

A limit is like walking halfway to a wall again and again: you never touch it, but you can say exactly where you're heading.

Intuition

Evaluate f at values closer and closer to a from both sides. If both sides approach the same number, that is the limit — even if f(a) itself is undefined (e.g. (x² − 4)/(x − 2) at x = 2).

Techniques

Direct substitution first. For 0/0: factor and cancel, rationalise surds, use standard limits or L'Hôpital. At infinity: compare dominant terms (exponentials beat polynomials, which beat logarithms).

One-sided and infinite limits

1/x → +∞ as x → 0⁺ and −∞ as x → 0⁻, so lim_{x→0} 1/x does not exist. Vertical asymptotes come from infinite limits; horizontal asymptotes from limits at infinity.

Formal definition and engineering use

The ε–δ definition makes 'arbitrarily close' precise. Engineers use limits for steady-state values (t → ∞), small-angle approximations (sin θ ≈ θ), and asymptotic behaviour of systems.

Formulas

• Existence: lim_{x→a} f(x) exists ⇔ left limit = right limit
• Standard limits: lim_{x→0} sin x / x = 1; lim_{x→0} (eˣ − 1)/x = 1; lim_{n→∞} (1 + 1/n)ⁿ = e
• Rational at infinity: Divide by the highest power of x
• L'Hôpital: 0/0 or ∞/∞: lim f/g = lim f′/g′
• ε–δ definition: ∀ε > 0 ∃δ > 0: 0 < |x − a| < δ ⇒ |f(x) − L| < ε

Worked example: Factor and cancel

Given: lim_{x→2} (x² − 4)/(x − 2)

Find: Limit.

• 0/0 at x = 2
• (x − 2)(x + 2)/(x − 2) = x + 2
• → 4

Answer: 4

Worked example: L'Hôpital

Given: lim_{x→0} (1 − cos x)/x²

Find: Limit.

• 0/0 → sin x/(2x)
• Still 0/0 → cos x/2
• → 1/2

Answer: 1/2

Common mistakes

• Wrong: Writing 0/0 = 1 or 0. — Right: 0/0 is indeterminate — simplify first.
• Wrong: Using L'Hôpital when not 0/0 or ∞/∞. — Right: Check the form first.

Applications

• Instantaneous rates: Velocity, current, heat flux as limits.
• Steady state: System output as t → ∞.
• Approximations: Small-angle and linear approximations.

Key takeaways

• Limits describe approach, not the value at a point.
• Left = right for a two-sided limit.
• Simplify 0/0 forms or use L'Hôpital.
• Limits at infinity give asymptotes and steady states.

Going further

• Squeeze theorem: Bound a function between two with equal limits.
• Asymptotic notation: O(), o(), ~ in analysis and algorithms.
• Sequences & series limits: Convergence tests.