LIMITS OF FUNCTIONS

RAHIM MOOSA

In this short note we discuss the notion of the limit of a function.

Given a function f : ℝ → ℝ and points p, q ∈ ℝ, we say that the limit of f at p is q, denoted by lim_{x→p} f(x) = q, to mean that for every ϵ > 0 there exists δ > 0 such

that |f(x) − q| < ϵ whenever 0 < |x − p| < δ. It is important here that we take |x − p| ≠ 0 in the above defiition. That is, whether or not q is the limit of f at p does not depend on the value of f at p.

As the following theorem shows, this can be rephrased in terms of sequences.

Theorem 0.1. Suppose f : ℝ → ℝ is a function and p, q ∈ ℝ. Then the following are equivalent:

(i) lim_{x→p} f(x) = q

(ii) For any sequence {p_{n}} with lim_{n→∞} p_{n} = p but p_{n} ≠ p, lim_{n→∞} f(p_{n}) = q.

Proof. Suppose (i) holds and let {p_{n}} be any sequence of real numbers converging 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. That is, (ii) holds.

Conversely, suppose (i) fails. 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 the sequence \{x(\frac{1}{n})\} witnesses the failure of (ii).