What height does a bouncing ball settle toward after many rebounds? That target height is never quite reached, yet it can be predicted. Mathematically, it is the value that a function approaches as time goes on. The language used to describe this idea is called limit notation.

Limits sit at the heart of derivatives and integrals, so the College Board dedicates two learning objectives to them—LIM-1.A.1 and LIM-1.B.1. Therefore, this lesson explains what a limit is, decodes the notation, presents three common representations, and highlights pitfalls that appear on both the AP® Calculus AB and BC exams.

What Exactly Is a Limit? 

Plain-language definition

A limit is the single number a function gets closer and closer to as its input gets closer and closer to some chosen point.

Formal statement

Using limit notation, this idea reads

\displaystyle \lim_{x\to c} f(x)=L

and is read “the limit of f of x as x approaches c equals L.”

Everyday analogy

Imagine walking toward a door but never stepping through it. Each step halves the remaining distance. Although the door is never reached, the walking position approaches a definite spot—the door’s threshold. That spot is the limit.

Step-by-step Example 1

Evaluate \displaystyle \lim_{x\to 3}(2x+1).

1. Substitute x = 3 directly because the expression is continuous.
   • 2(3)+1=7
2. Therefore, the function can be made arbitrarily close to 7 by taking x sufficiently close to 3.

Answer: 7

Quick Practice: Find \displaystyle \lim_{x\to-2}(5x-4).

Solution: Substitute to get 5(-2)-4=-14.

Cracking the Code: Limit Notation Explained

Anatomy of \displaystyle \lim_{x\to c} f(x)=L

• lim signals “take the limit.”
• The arrow x → c shows the approach value.
• f(x) is the function itself.
• L is the number being approached.

One-sided limits

Sometimes, only one side matters:

\lim_{x\to c^-}f(x) (approach from the left)

\lim_{x\to c^+}f(x) (approach from the right)

Infinite limits and limits at infinity

If f(x) grows without bound near c, write

\displaystyle \lim_{x\to c}f(x)=\infty.

If x itself heads to infinity, use

\displaystyle \lim_{x\to\infty}f(x)=L.

Step-by-step Example 2

Find \displaystyle \lim_{x\to0}\frac{\sin x}{x}.

1. Direct substitution gives 0⁄0, an indeterminate form.
2. On the unit circle, sin x is trapped between x and tan x. Therefore, by the Squeeze Theorem, \displaystyle \lim_{x\to0}\frac{\sin x}{x}=1.

Answer: 1

Mini-summary

• Limit notation precisely records the value that a function approaches.
• One-sided limits test direction.
• Infinite limits describe unbounded growth.

Three Ways to Represent a Limit

Graphical View

A graph reveals holes, jumps, and vertical asymptotes quickly.

Example 3 (graph description):

A piecewise curve follows y = x except at x = 2, where an open circle (hole) appears at (2, 2) and a solid dot is plotted at (2, 5). The y-value approached on both sides is 2, so

\displaystyle \lim_{x\to2}f(x)=2

…even though f(2)=5. This is a removable discontinuity.

Numerical (Table) View

Creating a table near c suggests the approaching value.

Example 4: Evaluate \displaystyle \lim_{x\to0}\frac{\sqrt{x+1}-1}{x} numerically.

x	–0.01	–0.001	0.001	0.01
f(x)	0.499	0.4999	0.5001	0.501

Numbers hover around 0.5, so the limit appears to be 0.5.

Analytical (Algebraic) View

Algebra often gives exact answers.

Common tactics

• Direct substitution
• Factoring & canceling
• Rationalizing
• Special trig limits

Example 5: Evaluate \displaystyle \lim_{x\to2}\frac{x^{2}-4}{x-2}.

1. Factor numerator: x^{2}-4=(x-2)(x+2).
2. Cancel the common factor: \frac{(x-2)(x+2)}{x-2}=x+2 for x ≠ 2.
3. Substitute: 2 + 2 = 4.

Answer: 4

Pros & Cons Quick List

• Graphical: easy to see behavior; however, scale may mislead.
• Numerical: useful when algebra is messy; nevertheless, rounding error exists.
• Analytical: provides exact value; yet, requires algebraic skill.

When Limits Do Not Exist

Some functions never settle on a single number.

1. Jump discontinuity – two one-sided limits differ.
2. Infinite limit – graph shoots upward or downward without bound.
3. Oscillation – function wiggles faster and faster.

Example 6: \displaystyle \lim_{x\to0}\frac{1}{x}

Approaching from the left gives –∞, while from the right gives ∞. Because the left and right limits disagree, the two-sided limit does not exist.

Why Limits Matter in AP® Calculus

Limits build directly into the formal derivative:

\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}.

Moreover, the Fundamental Theorem of Calculus relies on limit processes to connect areas with antiderivatives. On the AP® exam, roughly 10–12 % of multiple-choice questions and frequent FRQ parts require fluent use of limit notation. Therefore, practicing all three representations increases flexibility.

Quick Reference Chart: Key Vocabulary & Definitions

Term	Definition	Simple Example
Limit	The value that a function approaches as x nears a point	\lim_{x\to3}2x=6
Limit notation	Symbolic form \lim_{x\to c}f(x)=L	\lim_{x\to2}\frac{x^{2}-4}{x-2}=4
One-sided limit	Limit from only the right (⁺) or left (⁻)	\lim_{x\to1^-}f(x)
Removable discontinuity	Hole where limit exists but function value is missing or different	\lim_{x\to2}f(x)=2 while f(2)\neq2
Infinite limit	Function grows without bound near c	\lim_{x\to0}\frac{1}{x}=\infty
Limit at infinity	Value a function approaches as x→∞	\lim_{x\to\infty}\frac{1}{x}=0
Continuous function	Function with no breaks; limit equals value	\lim_{x\to5}(x^{2}+1)=26

Conclusion

A limit captures the single value that a function approaches as the input closes in on a target. Mastering limit notation—graphically, numerically, and analytically—builds a sturdy foundation for derivatives and integrals on both AP® Calculus exams. Next, explore derivative rules to see how limits power every slope formula encountered in calculus.

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