Limits and continuity

A limit describes the value a function gets closer and closer to as x approaches some number, even if the function isn’t defined exactly there. Limits are the foundation of calculus.

Key facts

  • lim_(x→a) f(x) is the value f(x) approaches as x gets close to a.
  • If substituting gives 0/0, factorise and cancel, then substitute.
  • As x → ∞, divide by the highest power of x.
  • Standard limits: sin x / x → 1 and (1 + 1/n)ⁿ → e.

Worked examples

Example 1

Find lim_(x→2) (x² − 4)/(x − 2).

  1. Substituting gives 0/0, so simplify first.
  2. (x − 2)(x + 2)/(x − 2) = x + 2
  3. As x → 2: 2 + 2 = 4

Answer: 4

Example 2

Find lim_(x→∞) (3x² + 1)/(x² − 5x).

  1. (3 + 1/x²)/(1 − 5/x)
  2. The small terms go to 0: 3/1 = 3

Answer: 3