Introduction to Limits

What Is a Limit?

Calculus is built on a single powerful idea: what happens to a function as its input gets arbitrarily close to some value? This is the concept of a limit, and it unlocks everything from derivatives to integrals.

Informally: the limit of f(x) as x approaches a is the value that f(x) gets closer and closer to as x gets closer and closer to a — from both sides.

The formal notation is:

$$\underset{x\to a}{\lim }f(x)=L$$

This reads: "the limit of f(x) as x approaches a equals L." Crucially, we're asking about what f(x) approaches, not what f(a) equals. The function might not even be defined at x = a, and the limit can still exist.

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Estimating Limits Graphically

The graph is your first tool. To find

$$\underset{x\to a}{\lim }f(x)$$

from a graph, trace the function from both the left and right sides of x = a and ask: where does it appear to be heading?

Three things can happen:

1. The function approaches the same value from both sides → the limit exists and equals that value
2. The function approaches different values from left and right → the limit does not exist (DNE)
3. The function increases or decreases without bound → the limit does not exist (though we write

   $$\underset{x\to a}{\lim }f(x)=\pm \infty$$

   to describe the behavior)

Key insight: A function can have a limit at x = a even if f(a) is undefined or has a different value. A hole in the graph (open circle) at x = a does not prevent the limit from existing — the surrounding behavior is what matters.

Interactive geometry — loading...

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One-Sided Limits

Sometimes it's useful to consider approach from only one direction.

• Left-hand limit:

  $$\underset{x\to a^{-}}{\lim }f(x)=L$$

  — x approaches a from values less than a
• Right-hand limit:

  $$\underset{x\to a^{+}}{\lim }f(x)=L$$

  — x approaches a from values greater than a

The two-sided limit exists if and only if both one-sided limits exist and are equal:

$$\underset{x\to a}{\lim }f(x)=L⟺\underset{x\to a^{-}}{\lim }f(x)=\underset{x\to a^{+}}{\lim }f(x)=L$$

If the one-sided limits disagree, the two-sided limit does not exist. This happens at jump discontinuities — common in piecewise functions.

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Estimating Limits Numerically

Before algebra gives us exact answers, tables give us strong numerical evidence. To estimate

$$\underset{x\to 2}{\lim }\frac{x^2-4}{x-2}$$

, build a table of values approaching x = 2 from both sides:

The values clearly approach 4, so we estimate

$$\underset{x\to 2}{\lim }\frac{x^2-4}{x-2}=4$$

. (At x = 2 itself, the function is undefined — 0/0 — but the limit exists.)

Caution: Numerical estimation can mislead. Values approaching a limit very slowly, or oscillating behavior near a point, can fool you. Tables are evidence, not proof. Use them to build intuition and check algebraic results.

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When Limits Do Not Exist

A limit fails to exist in three main scenarios:

1. Jump discontinuity: The left-hand and right-hand limits both exist but are unequal. Common with piecewise functions like

$$f(x)=\{\begin{array}{ll}1& \text{if }x<0\\ -1& \text{if }x\ge 0\end{array}$$

.

2. Infinite behavior: The function grows without bound near x = a. We write:

$$\underset{x\to 0^{+}}{\lim }\frac{1}{x}=+\infty$$

This describes the behavior but the limit technically does not exist (∞ is not a real number).

3. Oscillation: The function oscillates faster and faster without settling near any value. The classic example:

$$\underset{x\to 0}{\lim }\sin \left(\frac{1}{x}\right)\text{ does not exist}$$

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Concept Check

For the function f(x) = |x|/x, what are the left-hand and right-hand limits as x approaches 0? Does the two-sided limit exist?

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The left-hand limit is −1 (since |x|/x = −1 for x < 0) and the right-hand limit is +1 (since |x|/x = 1 for x > 0). Because these one-sided limits are unequal, the two-sided limit as x approaches 0 does not exist.

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The Formal Definition (Preview)

The intuitive notion of "approaching" can be made precise. The epsilon-delta definition states:

$$\underset{x\to a}{\lim }f(x)=L⟺\forall \epsilon >0,\exists \delta >0\text{ such that }0<|x-a|<\delta ⟹|f(x)-L|<\epsilon$$

In plain language: no matter how tight a tolerance ε you set around L, you can always find a neighborhood around a (within distance δ) where f(x) stays within that tolerance. AP Calculus AB does not require epsilon-delta proofs, but understanding the idea — that closeness of inputs guarantees closeness of outputs — is foundational.

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AP Exam Skills

No-calculator (Section I Part A): You must recognize limit notation, identify DNE situations, and read limits from graphs and tables without computational tools.

Calculator (Section I Part B): Use your calculator to evaluate functions at values close to a point and build numerical tables — but always interpret results analytically.

Free response: When you claim a limit does or does not exist, justify with evidence: cite specific one-sided limits, describe graph behavior, or reference the definition. "The limit equals 4" is not sufficient — explain why.

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Unit Summary

The limit is calculus's core idea: the value a function approaches as its input approaches a specific value, independent of what happens at that value. Key takeaways:

• Limits are about approach, not arrival
• Both one-sided limits must agree for a two-sided limit to exist
• Graphical and numerical estimation build intuition; algebra provides precision
• A function can have a limit at a point where it is undefined, equal to a different value, or completely undefined
• The epsilon-delta definition formalizes the notion of "getting arbitrarily close"

Next unit: We'll develop algebraic techniques — factoring, rationalization, conjugates — to evaluate limits exactly rather than estimating them.