2.5 The Precise Definition of a Limit

Limits are the foundation of calculus, describing how functions behave as they approach specific points. The epsilon-delta definition gives you a precise, airtight way to state what "approaching" actually means, using small intervals to pin down a function's behavior near a given value.

This rigorous approach is what allows mathematicians to prove limit properties rather than just assume them. Understanding the epsilon-delta definition gives you deeper insight into the ideas behind continuity, derivatives, and integrals.

The Precise Definition of a Limit

Epsilon-delta definition of limits

The intuitive idea of a limit says that $f(x)$ gets "closer and closer" to $L$ as $x$ gets "closer and closer" to $a$. The epsilon-delta definition makes that vague language precise by quantifying exactly what "closer" means.

The formal statement: $\lim_{x \to a} f(x) = L$ if and only if for every $\epsilon > 0$, there exists a $\delta > 0$ such that

$0 < |x - a| < \delta \implies |f(x) - L| < \epsilon$

Here's what each piece means:

• $\epsilon$ (epsilon) represents how close you want $f(x)$ to be to $L$. Think of it as a "tolerance" around the output. The expression $|f(x) - L| < \epsilon$ means $f(x)$ stays within $\epsilon$ units of $L$.
• $\delta$ (delta) represents how close $x$ needs to be to $a$ in order to guarantee that tolerance. The expression $0 < |x - a| < \delta$ means $x$ is within $\delta$ units of $a$, but not equal to $a$ itself.
• The condition $0 < |x - a|$ (strictly greater than zero) is important: the definition doesn't care what happens at $a$, only near $a$.

The key idea is that no matter how small someone makes $\epsilon$, you can always find a $\delta$ that works. The challenger picks $\epsilon$; your job is to produce $\delta$.

This definition eliminates ambiguity and makes rigorous proofs of limit properties possible. Absolute value is used throughout because it measures distance on the number line.

Application of the epsilon-delta approach

To prove a limit using the epsilon-delta definition, follow these steps:

1. Start with an arbitrary $\epsilon > 0$. You don't get to pick a specific number; the proof must work for every positive $\epsilon$.

2. Work backward from $|f(x) - L| < \epsilon$ to figure out what restriction on $|x - a|$ would guarantee it. This is your scratch work to find $\delta$.

3. Express $\delta$ in terms of $\epsilon$ so that whenever $0 < |x - a| < \delta$, the inequality $|f(x) - L| < \epsilon$ follows.

4. Write the formal proof, starting from "Let $\epsilon > 0$" and ending with the conclusion.

Example: Prove $\lim_{x \to 2} (3x - 1) = 5$.

Scratch work: You need $|(3x - 1) - 5| < \epsilon$. Simplify:

$|(3x - 1) - 5| = |3x - 6| = 3|x - 2|$

So you need $3|x - 2| < \epsilon$, which means $|x - 2| < \frac{\epsilon}{3}$. That tells you to choose $\delta = \frac{\epsilon}{3}$.

Formal proof:

• Let $\epsilon > 0$ be given.
• Choose $\delta = \frac{\epsilon}{3}$. Note $\delta > 0$.
• Suppose $0 < |x - 2| < \delta$. Then:

$|(3x - 1) - 5| = 3|x - 2| < 3 \cdot \frac{\epsilon}{3} = \epsilon$

Therefore, for every $\epsilon > 0$, there exists $\delta = \frac{\epsilon}{3} > 0$ such that $0 < |x - 2| < \delta \implies |(3x-1) - 5| < \epsilon$, which proves the limit.

The scratch work and the formal proof are separate stages. In the scratch work you figure out what $\delta$ should be; in the proof you verify it actually works.

One-sided and infinite limits

One-sided limits restrict $x$ to approach $a$ from only one direction. The definitions are the same as the standard epsilon-delta definition, except the $\delta$-condition changes:

• Left-hand limit: $\lim_{x \to a^-} f(x) = L$ means for every $\epsilon > 0$, there exists $\delta > 0$ such that

$a - \delta < x < a \implies |f(x) - L| < \epsilon$

• Right-hand limit: $\lim_{x \to a^+} f(x) = L$ means for every $\epsilon > 0$, there exists $\delta > 0$ such that

$a < x < a + \delta \implies |f(x) - L| < \epsilon$

For the two-sided limit $\lim_{x \to a} f(x) = L$ to exist, both one-sided limits must exist and be equal.

Infinite limits describe functions whose values grow without bound near a point. Instead of requiring $f(x)$ to stay within $\epsilon$ of some finite $L$, you require $f(x)$ to exceed any chosen bound $M$:

• Positive infinity: $\lim_{x \to a} f(x) = \infty$ means for every $M > 0$, there exists $\delta > 0$ such that

$0 < |x - a| < \delta \implies f(x) > M$

• Negative infinity: $\lim_{x \to a} f(x) = -\infty$ means for every $M < 0$, there exists $\delta > 0$ such that

$0 < |x - a| < \delta \implies f(x) < M$

Notice the structural parallel: $\epsilon$ gets replaced by $M$, and "close to $L$" gets replaced by "larger (or smaller) than $M$." The $\delta$ part works the same way.

Epsilon-delta support for limit laws

The epsilon-delta definition provides the rigorous foundation for the limit laws you use to evaluate limits of combined functions. Here are two key proofs that show how this works.

Sum Rule: If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, then $\lim_{x \to a} [f(x) + g(x)] = L + M$.

Proof outline:

• Let $\epsilon > 0$ be given.
• Since $\lim_{x \to a} f(x) = L$, there exists $\delta_1 > 0$ such that $0 < |x - a| < \delta_1 \implies |f(x) - L| < \frac{\epsilon}{2}$.
• Since $\lim_{x \to a} g(x) = M$, there exists $\delta_2 > 0$ such that $0 < |x - a| < \delta_2 \implies |g(x) - M| < \frac{\epsilon}{2}$.
• Choose $\delta = \min(\delta_1, \delta_2)$. Then for $0 < |x - a| < \delta$:

$|[f(x) + g(x)] - (L + M)| \leq |f(x) - L| + |g(x) - M| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$

The trick is splitting $\epsilon$ into two halves (one for each function) and using the triangle inequality ($|A + B| \leq |A| + |B|$) to combine them. Taking $\delta = \min(\delta_1, \delta_2)$ ensures both conditions hold simultaneously.

Constant Multiple Rule: If $\lim_{x \to a} f(x) = L$ and $c$ is a constant, then $\lim_{x \to a} [c \cdot f(x)] = c \cdot L$.

Proof outline (for $c \neq 0$):

• Let $\epsilon > 0$ be given.
• There exists $\delta > 0$ such that $0 < |x - a| < \delta \implies |f(x) - L| < \frac{\epsilon}{|c|}$.
• Then: $|c \cdot f(x) - c \cdot L| = |c| \cdot |f(x) - L| < |c| \cdot \frac{\epsilon}{|c|} = \epsilon$

(When $c = 0$, the result is trivial since $c \cdot f(x) = 0$ for all $x$.)

These proofs show the general strategy: manipulate the $\epsilon$ you're given to create the right conditions for each component, then combine the results.

Mathematical Foundations

A few background concepts that appear throughout epsilon-delta proofs:

• Functions map inputs to outputs. In calculus, you're typically working with functions from real numbers to real numbers.
• Real numbers include all rational and irrational numbers, represented as points on a number line. The completeness of the real numbers is what makes the epsilon-delta framework work.
• Absolute value $|x - a|$ measures the distance between $x$ and $a$ on the number line. This is why it shows up in every epsilon-delta statement.
• Inequalities express the relative size of two values. The entire epsilon-delta definition is built on inequalities that describe neighborhoods (intervals) around points.