LIMITS OF FUNCTIONS

Given f : ℝ → ℝ, lim_{x→p} f(x) = q means that |f(x)−q| < ϵ whenever 0 < |x−p| <

δ.

Theorem 1. lim_{x→p} f(x) = q if and only if lim_{n→∞} f(p_{n}) = q whenever lim_{n→∞} p_{n} = p

with p_{n} ≠ p.

Proof. Suppose lim_{x→p} f(x) = q and let {p_{n}} be any sequence of real numbers con-

verging to p, but not equal to p. Let ϵ > 0 be arbitrary. Then there exists a δ > 0 such that |f(x) − q| < ϵ if 0 < |x − p| < δ. On the other hand, there is a natural number N such that for all n > N, 0 < |p_{n} − p| < δ. Hence for all n > N, |f(p_{n}) − q| < ϵ, and so {f(p_{n})} converges to q.

Suppose lim_{x→p} f(x) ≠ q. Then there exists some ϵ > 0 such that for every δ > 0

there exists some x(δ) ∈ ℝ for which 0 < |x(δ)−p| < δ but |f(x(δ))−q| ≥ ϵ. Hence

lim_{n→∞} f(x(\frac{1}{n})) ≠ q but lim_{n→∞} x(\frac{1}{n}) = p.