The idea of a limit

The number a function's values settle towards, as x gets close to some point a, is its limit at a. We write lim_{x→a} f(x) = l.

For f(x) = x², the values shrink towards 0 as x approaches 0, so lim_{x→0} x² = 0. For g(x) = |x| too, the limit at 0 is 0.

The limit ignores the value at a itself. h(x) = (x² − 4)/(x − 2) is not defined at x = 2, yet for every other x it equals x + 2, so lim_{x→2} h(x) = 4.

Worked: lim_{x→5} (x + 10) = 15, lim_{x→1} x³ = 1, lim_{x→2} 3x = 6 and lim_{x→1} (x² + x) = 2. The limit of a constant function f(x) = 3 at any point is 3.

Worked: sin x at x = π/2 − 0.1 is about 0.9950 and at π/2 − 0.01 about 0.9999, so lim_{x→π/2} sin x = 1. Similarly lim_{x→0} (x + cos x) = 1.

A limit need not be a finite number. For f(x) = 1/x² with x > 0, the values grow without bound as x → 0, and we write lim f(x) = +∞.