4 Limits of Real Functions

In the preceding chapters, we studied sequences—functions from the natural numbers to the real line. We asked: given a_n, does there exist a real number a such that a_n approaches a as n grows large? The \varepsilon-N machinery formalized this intuition, and we proved fundamental theorems governing convergent sequences.

Yet sequences capture only discrete processes. Consider again the motivating examples from Section 2.1

A biologist records bacterial population each day: P_1, P_2, P_3, \dots These measurements form a sequence; however, the underlying bacterial growth is continuous—between measurements at t = 5 and t = 6, the population evolves smoothly.
A physicist measures a particle’s position at integer times t = 0, 1, 2, \dots, producing a sequence s_0, s_1, s_2, \dots But the particle moves through all intermediate positions between these snapshots.
A patient’s blood pressure is monitored weekly. The measurements form a sequence, yet cardiovascular function is a continuous physiological process.

In each case, the discrete sequence is an approximation of an underlying continuous phenomenon. To model reality, we must study functions defined on continuous domains—intervals of real numbers rather than isolated integers.

This chapter extends the limit concept from sequences to functions f : \mathbb{R} \to \mathbb{R}. The guiding question shifts

Sequences: Does a_n approach a as n \to \infty?
Functions: Does f(x) approach L as x \to a?

The mathematical framework remains nearly identical. Where we previously required |a_n - a| < \varepsilon for all sufficiently large n, we now require |f(x) - L| < \varepsilon for all x sufficiently close to a. The discrete threshold N is replaced by a continuous tolerance \delta. This parallels the transition from discrete measurements to continuous observation.

4.1 Motivation: Instantaneous Velocity

To ground the notion of a limit, consider a concrete physical problem: determining the instantaneous velocity of a moving object.

Suppose a particle moves along a line, and its position at time t is given by s(t). The average velocity over the interval [t_0, t_0 + h] is

\bar{v} = \frac{s(t_0 + h) - s(t_0)}{h}.

This is the displacement divided by elapsed time—familiar from elementary physics.

But what is the particle’s velocity at the instant t = t_0? Setting h = 0 yields the meaningless expression \frac{0}{0}. Yet intuitively, we expect the particle to possess a well-defined velocity at each moment.

The resolution: examine what happens as h becomes arbitrarily small. If the average velocity v_{\text{avg}} approaches a definite value as h \to 0, we declare that value to be the instantaneous velocity at t_0. Formally,

v(t_0) = \lim_{h \to 0} \frac{s(t_0 + h) - s(t_0)}{h},

provided this limit exists.

This is not a sequence limit—h varies continuously through all real numbers near zero, not just integers. We require a new definition.

4.2 The Limit of a Function

We now formalize the notion of a function approaching a limit.

Definition 4.1 (Limit of a Function (Informal)) Let f be a function defined on an open interval containing a, except possibly at a itself. We say that

\lim_{x \to a} f(x) = L

if f(x) can be made arbitrarily close to L by taking x sufficiently close to a (but not equal to a).

This informal definition captures the intuition: as x approaches a, the values f(x) cluster around L. Crucially, the value f(a) itself is irrelevant—f need not even be defined at a.

4.2.0.1 Example

Consider

f(x) = \frac{x^2 - 1}{x - 1}.

At x = 1, the denominator vanishes, so f(1) is undefined. Yet for x \neq 1,

f(x) = \frac{(x-1)(x+1)}{x-1} = x + 1.

As x approaches 1, f(x) approaches 2. We write \lim_{x \to 1} f(x) = 2, despite f(1) being undefined.

4.3 The \varepsilon-\delta Definition

The informal definition suffices for intuition, but in mathematics we demand precision. What does “arbitrarily close” and “sufficiently close” mean quantitatively?

Recall the \varepsilon-N definition of sequence convergence: for every tolerance \varepsilon > 0, there exists a threshold N such that all terms beyond N lie within \varepsilon of the limit.

For functions, we replace the integer threshold N with a distance threshold \delta > 0. The value f(x) must lie within \varepsilon of L whenever x lies within \delta of a (excluding a itself).

Definition 4.2 (Limit of a Function) 0 < |x-a| < \delta, \ x \in U \implies |f(x)-L| < \varepsilon.

Note: The statment |x - a| < \delta means x lies in the interval (a - \delta, a + \delta). While 0 < |x - a| excludes x = a itself. The function’s value at a is not relevant; only its behavior near a matters. Finally, |f(x) - L| < \varepsilon means f(x) lies in the interval (L - \varepsilon, L + \varepsilon).

The definition asserts: no matter how tight the tolerance \varepsilon, we can find a neighborhood around a (of radius \delta) such that all f(x) values within that neighborhood (excluding a) are within \varepsilon of L.

Example 4.1 (Limit of a Linear Function) Let f(x) = 2x + 3. We show that \lim_{x \to 1} f(x) = 5.

Proof. Let \varepsilon > 0. Set \delta = \varepsilon/2. Then for 0 < |x - 1| < \delta,

|f(x) - 5| = |2x + 3 - 5| = 2|x - 1| < 2 \cdot \frac{\varepsilon}{2} = \varepsilon.

Hence, \lim_{x \to 1} f(x) = 5. \square

Scratch Work

We want 0 < |x - 1| < \delta \implies |f(x) - 5| < \varepsilon.

|f(x) - 5| = |2x + 3 - 5| = 2|x - 1|.

Require 2|x - 1| < \varepsilon, hence

|x - 1| < \frac{\varepsilon}{2}.

This motivates our choice

\delta = \frac{\varepsilon}{2}.

Interpretation

For a linear function, the “response” |f(x) - L| is proportional to the “perturbation” |x - a|. The proportionality constant (here, 2) determines how tightly \delta must be chosen relative to \varepsilon. That is, smaller tolerances \varepsilon require proportionally smaller neighborhoods \delta around a.

It is useful to relate the \varepsilon–\delta definition of a limit to the behavior of a function along sequences. The idea of letting x approach p is naturally expressed by taking a sequence of points that converges to p, and many intuitive arguments in elementary calculus proceed in this way even if the language of sequences is not mentioned. The following theorem makes this connection precise. It shows that the limit of a function at a point is completely determined by the values of the function on all sequences converging to that point.

Theorem 4.1 (Sequential Criterion for Limits) Let E\subset\mathbb{R}, f:E\to\mathbb{R}, and let p be a limit point of E. The following two statements are equivalent

\lim_{x\to p} f(x)=q.
For every sequence (p_n) in E with p_n\neq p for all n and p_n\to p, we have f(p_n)\to q.

Proof.

(\Rightarrow) Assume first that \lim_{x\to p} f(x)=q. Fix \varepsilon>0. By the definition of the limit, there is a \delta>0 such that |f(x)-q|<\varepsilon whenever x\in E and 0<|x-p|<\delta. If (p_n) is any sequence in E with p_n\neq p and p_n\to p, then there is N such that |p_n-p|<\delta for all n>N. For such n we have |f(p_n)-q|<\varepsilon, so f(p_n)\to q.

It is natural to ask whether a function can approach two distinct values at the same point. By analogy with sequences, whose limits are unique (see Theorem 3.1), one expects the same for functions. Indeed, if \lim_{x \to a} f(x) exists, it is unique. No function can converge to two different values at the same point; any assumption to the contrary immediately contradicts the uniqueness of sequence limits. This fundamental fact underlies all further considerations of limits.

Theorem 4.2 (Uniqueness of Limits of Functions) Let f: U \to \mathbb{R} with U \subset \mathbb{R}, and let a be a limit point of U. If \lim_{x \to a} f(x) = L_1 \quad \text{and} \quad \lim_{x \to a} f(x) = L_2, then L_1 = L_2.

Proof.

Suppose, for contradiction, that L_1 \neq L_2, and set \varepsilon = |L_1 - L_2|/2 > 0. By definition, there exist \delta_1, \delta_2 > 0 such that

0 < |x - a| < \delta_1 \implies |f(x) - L_1| < \varepsilon, \quad 0 < |x - a| < \delta_2 \implies |f(x) - L_2| < \varepsilon.

Let \delta = \min(\delta_1, \delta_2). For any x with 0 < |x - a| < \delta, both inequalities hold, so by the triangle inequality,

|L_1 - L_2| \le |f(x) - L_1| + |f(x) - L_2| < 2 \varepsilon = |L_1 - L_2|,

a contradiction. Hence L_1 = L_2. \square

4.4 One-Sided Limits

Sometimes a function behaves differently as x approaches a from the left versus from the right. We introduce one-sided limits to capture this distinction.

Definition 4.3 (One-Sided Limits) We say

\lim_{x \to a^-} f(x) = L

if for every \varepsilon > 0 there exists \delta > 0 such that

a - \delta < x < a \implies |f(x) - L| < \varepsilon.

Similarly,

\lim_{x \to a^+} f(x) = L

if for every \varepsilon > 0 there exists \delta > 0 such that

a < x < a + \delta \implies |f(x) - L| < \varepsilon.

Theorem 4.3 (Two-Sided Limit via One-Sided Limits) If both one-sided limits of f at c exist and are equal, then the two-sided limit exists and equals this common value.

Proof.

Let L be the common value. Given \varepsilon > 0, choose \delta^+ > 0 and \delta^- > 0 such that

|f(x) - L| < \varepsilon \quad \text{whenever } 0 < x - c < \delta^+ \text{ or } 0 < c - x < \delta^-.

Set

\delta = \min(\delta^-, \delta^+).

Then for all x with 0 < |x - c| < \delta, x lies on one side of c, so |f(x) - L| < \varepsilon. Hence f(x) \to L as x \to c. \square

Example 4.2 (The Heaviside Step Function) Define

H(x) = \begin{cases} 0 & \text{if } x < 0, \\ 1 & \text{if } x \ge 0. \end{cases}

At x = 0

\lim_{x \to 0^-} H(x) = 0 (approaching from the left, H(x) = 0)
\lim_{x \to 0^+} H(x) = 1 (approaching from the right, H(x) = 1)

Since the one-sided limits differ, \lim_{x \to 0} H(x) does not exist.