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This lecture formalises the concept of limits — the mathematical foundation upon which all of calculus is built — and introduces techniques for evaluating them.

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📋 At a Glance

• [ ] Understand the intuitive definition of a limit
• [ ] Evaluate limits using direct substitution
• [ ] Apply algebraic techniques when direct substitution fails
• [ ] Evaluate limits involving indeterminate forms (0/0)
• [ ] Understand one-sided limits (left and right)
• [ ] Apply the Squeeze Theorem
• [ ] Evaluate limits at infinity
• [ ] Recognise when limits do not exist

1. The Concept of a Limit

Intuitive Definition

We write:

lim_{x→a} f(x) = L

if f(x) can be made arbitrarily close to L by making x sufficiently close to a (but x ≠ a).

Key Point

The limit describes the behaviour near the point, NOT at the point. The function doesn't even need to be defined at x = a.

Example