The Epsilon-Delta Definition of a Limit

MATH161

Reference: Stewart 2.4 • Chapter: 1 • Section: 4

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The Epsilon-Delta Definition of a Limit

Why Make Limits Precise?

You already know intuitively what $\lim_{x \to a} f(x) = L$ means: as $x$ gets closer to $a$, $f(x)$ gets closer to $L$. But "closer" is vague. How close is close enough? Can we guarantee that $f(x)$ stays within any desired distance of $L$?

The epsilon-delta definition answers this question with mathematical precision. It transforms "closer and closer" into a concrete promise: no matter how tight a tolerance you demand for the output, I can always find a corresponding tolerance for the input that makes it work.

This definition underlies everything in calculus—continuity, derivatives, and integrals all depend on it. Understanding it gives you the power to prove that limits exist rather than just guess from graphs.

Prerequisite Map

Quick Reference

Property	Value
Course	MATH161
Chapter.Section	1.7
Difficulty	Intermediate
Time	~15 minutes

Key Concepts

The Formal Definition

Definition. We say $\lim_{x \to a} f(x) = L$ if and only if:

$$\forall\, \varepsilon > 0,\ \exists\, \delta > 0 \text{ such that } 0 < \vert x - a\vert < \delta \Rightarrow \vert f(x) - L\vert < \varepsilon$$

Let's unpack each part:

Symbol	Meaning	Role
$\varepsilon > 0$	Any positive tolerance for the output	How close $f(x)$ must be to $L$
$\delta > 0$	A positive tolerance for the input	How close $x$ must be to $a$
$0 < \\|x - a\\|$	$x \neq a$ (but close to $a$)	We don't care about $f(a)$ itself
$\\|x - a\\| < \delta$	$x$ is within $\delta$ of $a$	The input condition
$\\|f(x) - L\\| < \varepsilon$	$f(x)$ is within $\varepsilon$ of $L$	The output guarantee

The key insight: For every $\varepsilon$ (no matter how small), there exists a $\delta$ that works. The challenger picks $\varepsilon$; you must find $\delta$.

The Interval Interpretation

The definition can be restated using intervals:

$$x \in (a - \delta, a + \delta) \setminus \{a\} \quad\Rightarrow\quad f(x) \in (L - \varepsilon, L + \varepsilon)$$

Input interval: $(a - \delta, a + \delta)$, excluding $a$ itself (a "$\delta$-neighborhood")
Output interval: $(L - \varepsilon, L + \varepsilon)$ (an "$\varepsilon$-neighborhood")

Graphical Interpretation

    y
    │
L+ε ├─────────────────────────────────────  ← upper ε-bound
    │                 ████████
    │              ███        ███
  L ├─────────────█──────────────█────────  ← limit value
    │           ██                ██
L-ε ├──────────█────────────────────█─────  ← lower ε-bound
    │         █                      █
    │        █                        █
    └────────┼──────────┼──────────┼──────→ x
           a-δ         a          a+δ

           ◄────────────────────────►
              δ-neighborhood of a

Reading the picture: Draw horizontal lines at $y = L + \varepsilon$ and $y = L - \varepsilon$. The definition guarantees you can find a vertical strip (the $\delta$-neighborhood) where the graph stays entirely between the horizontal lines.

Why We Exclude $x = a$

The condition $0 < \vert x - a\vert $ means $x \neq a$. This is essential because:

$f(a)$ might not exist (e.g., $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$)
$f(a)$ might not equal $L$ (e.g., a removable discontinuity)
Limits describe behavior approaching $a$, not at $a$

The limit is entirely about what happens near $a$, not at $a$.

Practice Problems

Level 1 Translating the Definition

Write the epsilon-delta definition for $\lim_{x \to 3} f(x) = 7$ by filling in the specific values.

Thought Process

Start with the template: "For every $\varepsilon > 0$, there exists $\delta > 0$ such that..."

Then substitute $a = 3$ and $L = 7$ into $\vert x - a\vert < \delta$ and $\vert f(x) - L\vert < \varepsilon$.

Show Answer

For every $\varepsilon > 0$, there exists $\delta > 0$ such that:

$$0 < \vert x - 3\vert < \delta \quad\Rightarrow\quad \vert f(x) - 7\vert < \varepsilon$$

In interval form: if $x \in (3 - \delta, 3 + \delta) \setminus \{3\}$, then $f(x) \in (7 - \varepsilon, 7 + \varepsilon)$.

Level 2 From Intervals to Inequalities

Convert the following statement to epsilon-delta notation:

"For the limit $\lim_{x \to 2} g(x) = 5$, when $x$ is within 0.1 of 2 (but not equal to 2), $g(x)$ is within 0.03 of 5."

Thought Process

Identify the pieces:

"$x$ is within 0.1 of 2" means $\vert x - 2\vert < 0.1$, so $\delta = 0.1$
"$g(x)$ is within 0.03 of 5" means $\vert g(x) - 5\vert < 0.03$, so $\varepsilon = 0.03$
"but not equal to 2" is the condition $x \neq 2$, which gives $0 < \vert x - 2\vert $

Show Answer

The statement says: when $\varepsilon = 0.03$, we can choose $\delta = 0.1$:

$$0 < \vert x - 2\vert < 0.1 \quad\Rightarrow\quad \vert g(x) - 5\vert < 0.03$$

This demonstrates one instance of the epsilon-delta condition. For the limit to truly equal 5, such a $\delta$ must exist for every $\varepsilon > 0$.

Level 3 Graphical Interpretation

A function $f$ has $\lim_{x \to 4} f(x) = 2$. On a graph:

If you draw horizontal lines at $y = 1.9$ and $y = 2.1$, what tolerance $\varepsilon$ does this represent?
A student draws vertical lines at $x = 3.7$ and $x = 4.3$, claiming this $\delta$ works. What is this $\delta$?
What must be true about the graph between $x = 3.7$ and $x = 4.3$ (excluding $x = 4$) for the student's claim to be correct?

Thought Process

For (a): The horizontal lines are at $L \pm \varepsilon$, so find $\varepsilon$ from $2.1 - 2$ or $2 - 1.9$.

For (b): The vertical lines are at $a \pm \delta$, so find $\delta$ from $4.3 - 4$ or $4 - 3.7$.

For (c): The graph must stay inside the horizontal band whenever $x$ is in the vertical strip.

Show Answer

$\varepsilon = 0.1$ (since $2.1 - 2 = 0.1$ and $2 - 1.9 = 0.1$)
$\delta = 0.3$ (since $4.3 - 4 = 0.3$ and $4 - 3.7 = 0.3$)
For every $x$ with $3.7 < x < 4.3$ and $x \neq 4$, we need $1.9 < f(x) < 2.1$. In other words, the entire graph (excluding possibly the point at $x = 4$) must lie within the horizontal band.

Level 4 Why $f(a)$ Doesn't Matter

Consider the function: $$h(x) = \begin{cases} x + 1 & \text{if } x \neq 2 \\ 100 & \text{if } x = 2 \end{cases}$$

What is $h(2)$?
Use the epsilon-delta definition to argue that $\lim_{x \to 2} h(x) = 3$ (you don't need to find $\delta$ explicitly, just explain why such a $\delta$ exists).
Why doesn't $h(2) = 100$ contradict the limit being 3?

Thought Process

The definition says $0 < \vert x - 2\vert $, which means $x \neq 2$. So when checking whether $\vert h(x) - 3\vert < \varepsilon$, we only look at values where $x \neq 2$, where $h(x) = x + 1$.

For all $x \neq 2$: $\vert h(x) - 3\vert = \vert (x + 1) - 3\vert = \vert x - 2\vert $. This equals exactly the distance from $x$ to 2.

Show Answer

$h(2) = 100$
For $x \neq 2$, we have $h(x) = x + 1$, so $\vert h(x) - 3\vert = \vert x + 1 - 3\vert = \vert x - 2\vert $.
Given any $\varepsilon > 0$, choose $\delta = \varepsilon$. Then: $$0 < \vert x - 2\vert < \delta = \varepsilon \quad\Rightarrow\quad \vert h(x) - 3\vert = \vert x - 2\vert < \varepsilon$$

So $\lim_{x \to 2} h(x) = 3$.</li>
The condition $0 < \vert x - 2\vert $ explicitly excludes $x = 2$. The limit only considers what happens as $x$ approaches 2, not what happens at $x = 2$. The value $h(2) = 100$ is irrelevant to the limit.

Level 5 Negating the Definition

The negation of the limit definition can be used to prove a limit does not exist.

Write the logical negation of: "For every $\varepsilon > 0$, there exists $\delta > 0$ such that $0 < \vert x - a\vert < \delta \Rightarrow \vert f(x) - L\vert < \varepsilon$."
Use your negation to explain why $\lim_{x \to 0} \frac{\vert x\vert }{x}$ is not equal to 1.
Can this limit equal any real number $L$? Explain.

Thought Process

For (a): Negate quantifiers: $\forall$ becomes $\exists$, and $\exists$ becomes $\forall$. The implication $P \Rightarrow Q$ negates to $P \land \neg Q$.

For (b) and (c): Notice that $\frac{\vert x\vert }{x} = 1$ for $x > 0$ and $\frac{\vert x\vert }{x} = -1$ for $x < 0$. Any $\delta$-neighborhood of 0 contains both positive and negative values.

Show Answer

The negation is: **There exists** $\varepsilon > 0$ such that **for all** $\delta > 0$, there exists $x$ with $0 < \vert x - a\vert < \delta$ **and** $\vert f(x) - L\vert \geq \varepsilon$.
In words: there's some tolerance $\varepsilon$ that cannot be achieved no matter how small we make $\delta$.</li>
Let $f(x) = \frac{\vert x\vert }{x}$. Note that $f(x) = 1$ for $x > 0$ and $f(x) = -1$ for $x < 0$.
Choose $\varepsilon = 1$. For any $\delta > 0$, the interval $(0 - \delta, 0 + \delta) \setminus \{0\}$ contains negative numbers like $x = -\delta/2$.

For such $x$: $\vert f(x) - 1\vert = \vert -1 - 1\vert = 2 \geq 1 = \varepsilon$.

So there exists $\varepsilon = 1$ for which no $\delta$ works. Therefore $\lim_{x \to 0} \frac{\vert x\vert }{x} \neq 1$.</li>
The limit cannot equal any $L$. Choose $\varepsilon = 0.5$.
- If $L \geq 0$: Any $\delta$-neighborhood contains $x < 0$ where $f(x) = -1$, so $\vert f(x) - L\vert = \vert -1 - L\vert \geq 1 > 0.5$.
- If $L < 0$: Any $\delta$-neighborhood contains $x > 0$ where $f(x) = 1$, so $\vert f(x) - L\vert = \vert 1 - L\vert > 1 > 0.5$.
In both cases, the $\varepsilon = 0.5$ challenge cannot be met, so no limit exists.</li>

Conceptual Check (CCI-Style)

Question 1: Which statement best describes what "$\lim_{x \to 5} f(x) = 12$" means?

$f(5) = 12$
As $x$ gets sufficiently close to 5, $f(x)$ can be made as close to 12 as desired
$f(x)$ approaches but never equals 12
There is some point near $x = 5$ where $f(x) = 12$

Answer

(B) is correct. The epsilon-delta definition says that for any desired closeness to 12 (any $\varepsilon$), we can guarantee $f(x)$ achieves that closeness by making $x$ sufficiently close to 5 (choosing appropriate $\delta$).

(A) is wrong because $f(5)$ need not equal 12 (or even exist). (C) is wrong because $f(x)$ can equal 12 (the definition doesn't forbid this). (D) is wrong because we need $f(x)$ close to 12 for all $x$ near 5, not just at some point.

Question 2: If $\lim_{x \to a} f(x) = L$, and someone challenges you with $\varepsilon = 0.001$, what are you claiming you can do?

Answer

You claim you can find a $\delta > 0$ such that whenever $x$ is within $\delta$ of $a$ (but not equal to $a$), the value $f(x)$ will be within 0.001 of $L$.

The key point: no matter how small the challenger makes $\varepsilon$, you can always find a working $\delta$.

Mastery Checklist

[ ] I can state the epsilon-delta definition of $\lim_{x \to a} f(x) = L$
[ ] I can explain each symbol ($\varepsilon$, $\delta$, $a$, $L$) and its role
[ ] I can translate between inequality form and interval form of the definition
[ ] I can interpret the definition graphically using horizontal and vertical bands
[ ] I can explain why we exclude $x = a$ from consideration
[ ] I can explain why $f(a)$ is irrelevant to the limit
[ ] I can write the negation of the definition to show a limit doesn't exist

Mental Model

The Epsilon-Delta Game:

Think of epsilon-delta as a challenge-response game:

Challenger picks $\varepsilon$: "I demand $f(x)$ be within $\varepsilon$ of $L$!"
You respond with $\delta$: "Easy—just keep $x$ within $\delta$ of $a$."
The limit exists if you can always win, no matter how small $\varepsilon$ is.

The smaller the challenger makes $\varepsilon$, the smaller you might need to make $\delta$—but you always have a winning response.

Connections

Looking back:

Limit Intuition gave us the informal idea; now we have the rigorous definition.
Absolute value as distance: $\vert x - a\vert $ measures how far $x$ is from $a$.

Looking ahead:

Epsilon-Delta Proofs will show how to find $\delta$ for specific functions.
The same $\varepsilon$-$\delta$ framework defines continuity: $f$ is continuous at $a$ when $\lim_{x \to a} f(x) = f(a)$.

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Limit Laws	Skills Index	Epsilon-Delta Proofs: Linear

Last updated: 2026-01-22