Unit 1.2 – Defining Limits and Using Limit Notation

AP® Calculus AB & BC | Formula Reference Sheet

Core Concept: A limit describes the value that a function \(f(x)\) approaches as \(x\) gets arbitrarily close to a specific value \(a\). The limit is NOT necessarily equal to \(f(a)\) — it describes behavior near the point, not at the point.

📝 Basic Limit Notation

Two-Sided Limit (Standard Limit)

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

Read as: "The limit of \(f(x)\) as \(x\) approaches \(a\) equals \(L\)"

Meaning: As \(x\) gets arbitrarily close to \(a\) (from both sides), the function \(f(x)\) gets arbitrarily close to \(L\).

📖 How to Read Limit Notation:

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

\(\lim\) = "limit" (the limiting value)
\(x \to a\) = "as \(x\) approaches \(a\)" (the input gets close to \(a\))
\(f(x)\) = the function we're examining
\(= L\) = the limit value (what \(f(x)\) approaches)

⚠️ Critical Point: The limit \(\lim_{x \to a} f(x)\) depends only on values of \(x\) near \(a\), NOT on \(f(a)\) itself. The function doesn't even need to be defined at \(x = a\) for the limit to exist!

⬅️➡️ One-Sided Limits

Left-Hand Limit

Left-Hand Limit (Approaching from the Left)

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

Read as: "The limit of \(f(x)\) as \(x\) approaches \(a\) from the left equals \(L\)"

Meaning: As \(x\) approaches \(a\) from values less than \(a\) (i.e., \(x < a\)), \(f(x)\) approaches \(L\).

Notation: The superscript "−" indicates approaching from the negative side (left).

Right-Hand Limit

Right-Hand Limit (Approaching from the Right)

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

Read as: "The limit of \(f(x)\) as \(x\) approaches \(a\) from the right equals \(L\)"

Meaning: As \(x\) approaches \(a\) from values greater than \(a\) (i.e., \(x > a\)), \(f(x)\) approaches \(L\).

Notation: The superscript "+" indicates approaching from the positive side (right).

✅ When Does a Limit Exist?

Condition for Limit Existence

A two-sided limit exists if and only if:

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

That is, both one-sided limits exist AND are equal. When this condition is met:

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

💡 Key Insight: If the left-hand and right-hand limits differ, the two-sided limit does not exist (DNE). This indicates a jump discontinuity.

❌ When Does a Limit NOT Exist (DNE)?

A limit does not exist when:

Left and Right Limits Differ: \(\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)\) (Jump Discontinuity)
Infinite Limit: \(f(x)\) grows without bound (approaches \(\infty\) or \(-\infty\)) (Vertical Asymptote)
Oscillation: \(f(x)\) oscillates wildly and doesn't settle to a single value (e.g., \(\sin\left(\frac{1}{x}\right)\) as \(x \to 0\))

Notation: Write \(\lim_{x \to a} f(x)\) does not exist or DNE

♾️ Limits at Infinity (Horizontal Asymptotes)

Limit as x Approaches Positive Infinity

\[ \lim_{x \to \infty} f(x) = L \]

Meaning: As \(x\) grows without bound (becomes very large), \(f(x)\) approaches \(L\). This describes a horizontal asymptote at \(y = L\).

Limit as x Approaches Negative Infinity

\[ \lim_{x \to -\infty} f(x) = L \]

Meaning: As \(x\) becomes very negative (approaches negative infinity), \(f(x)\) approaches \(L\).

📝 Note: If \(\lim_{x \to \infty} f(x) = L\) or \(\lim_{x \to -\infty} f(x) = L\), then \(y = L\) is a horizontal asymptote.

📈 Infinite Limits (Vertical Asymptotes)

Infinite Limit (Positive Infinity)

\[ \lim_{x \to a} f(x) = \infty \]

Meaning: As \(x\) approaches \(a\), \(f(x)\) increases without bound. This indicates a vertical asymptote at \(x = a\).

Infinite Limit (Negative Infinity)

\[ \lim_{x \to a} f(x) = -\infty \]

Meaning: As \(x\) approaches \(a\), \(f(x)\) decreases without bound (goes to negative infinity).

⚠️ Important: When \(\lim_{x \to a} f(x) = \pm\infty\), we say the limit does not exist in the finite sense, but we can still describe the behavior as "approaching infinity." This creates a vertical asymptote at \(x = a\).

One-Sided Infinite Limits

\[ \lim_{x \to a^-} f(x) = \infty \quad \text{or} \quad \lim_{x \to a^+} f(x) = -\infty \]

One-sided limits can also be infinite, indicating the direction from which the function approaches the vertical asymptote.

📋 Complete Limit Notation Summary

Notation	Read As	Meaning
\(\lim_{x \to a} f(x) = L\)	Limit as x approaches a	\(f(x)\) approaches \(L\) from both sides
\(\lim_{x \to a^-} f(x) = L\)	Left-hand limit	\(f(x)\) approaches \(L\) from the left (\(x < a\))
\(\lim_{x \to a^+} f(x) = L\)	Right-hand limit	\(f(x)\) approaches \(L\) from the right (\(x > a\))
\(\lim_{x \to \infty} f(x) = L\)	Limit at positive infinity	Horizontal asymptote \(y = L\) as \(x \to \infty\)
\(\lim_{x \to -\infty} f(x) = L\)	Limit at negative infinity	Horizontal asymptote \(y = L\) as \(x \to -\infty\)
\(\lim_{x \to a} f(x) = \infty\)	Infinite limit	Vertical asymptote at \(x = a\) (unbounded growth)
\(\lim_{x \to a} f(x) = -\infty\)	Infinite limit (negative)	Vertical asymptote at \(x = a\) (unbounded decrease)
\(\lim_{x \to a} f(x)\) DNE	Limit does not exist	Left/right limits differ, or oscillation occurs

🔢 Three Ways to Represent Limits

Limits can be represented in three ways:

Analytically (Algebraically): Using formulas and limit notation
Numerically: Using tables of values approaching the target
Graphically: Observing the behavior on a graph

1. Analytical Representation

Using algebraic manipulation and limit laws to evaluate:

\[ \lim_{x \to 3} (2x + 5) = 2(3) + 5 = 11 \]

2. Numerical Representation

Create a table with \(x\) values approaching the target from both sides:

\(x\) (from left)	\(f(x)\)	\(x\) (from right)	\(f(x)\)
2.9	10.8	3.1	11.2
2.99	10.98	3.01	11.02
2.999	10.998	3.001	11.002

Both sides approach 11, so \(\lim_{x \to 3} f(x) = 11\)

3. Graphical Representation

Observe the y-values as the graph approaches \(x = a\) from both sides. The limit is the y-value the function approaches, not necessarily the actual y-value at that point.

🔍 Limit vs. Function Value

Critical Distinction:

Concept	Symbol	What It Represents
Function Value	\(f(a)\)	The actual value of the function at \(x = a\)
Limit	\(\lim_{x \to a} f(x)\)	The value \(f(x)\) approaches as \(x\) gets close to \(a\)

📝 Key Point: \(\lim_{x \to a} f(x)\) can exist even if:

\(f(a)\) is undefined (removable discontinuity/hole)
\(f(a)\) has a different value than the limit
The function has a break or gap at \(x = a\)

📚 Important Examples

Example 1: Limit Exists, Function Undefined

\[ \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \]

Step 1: Factor the numerator: \(\frac{(x-2)(x+2)}{x-2}\)

Step 2: Cancel common factors (for \(x \neq 2\)): \(x + 2\)

Step 3: Evaluate: \(\lim_{x \to 2} (x+2) = 2 + 2 = 4\)

Note: \(f(2)\) is undefined (\(\frac{0}{0}\)), but \(\lim_{x \to 2} f(x) = 4\)

Example 2: One-Sided Limits Differ

Consider: \(f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ 2x & \text{if } x \geq 1 \end{cases}\)

\[ \lim_{x \to 1^-} f(x) = 1^2 = 1 \]

\[ \lim_{x \to 1^+} f(x) = 2(1) = 2 \]

Since \(1 \neq 2\), \(\lim_{x \to 1} f(x)\) does not exist (DNE).

Example 3: Limit at Infinity

\[ \lim_{x \to \infty} \frac{3x^2 + 2}{5x^2 + 4x + 1} \]

Step 1: Divide numerator and denominator by highest power (\(x^2\)):

\[ \frac{3 + \frac{2}{x^2}}{5 + \frac{4}{x} + \frac{1}{x^2}} \]

Step 2: As \(x \to \infty\), terms with \(x\) in denominator approach 0:

\[ \lim_{x \to \infty} \frac{3 + 0}{5 + 0 + 0} = \frac{3}{5} \]

💡 Tips, Tricks & Common Mistakes

✅ Essential Tips

Arrow vs. Equals: Use \(x \to a\) (arrow) for "approaching," not \(x = a\) (equals)
One-Sided Check: Always check both left and right limits for piecewise functions
DNE vs. Infinity: \(\lim = \infty\) is technically DNE, but we describe it as "approaching infinity"
Removable Discontinuity: If \(\lim_{x \to a} f(x) = L\) exists but \(f(a) \neq L\) or undefined, there's a hole
Table Method: Use symmetric values (e.g., 2.9, 2.99 and 3.1, 3.01) for better accuracy

🎯 Math Tricks & Shortcuts

Trick 1: For limits at infinity with rational functions:

If degree of numerator = degree of denominator: Limit = ratio of leading coefficients
If degree of numerator < degree of denominator: Limit = 0
If degree of numerator > degree of denominator: Limit = ±∞

Trick 2: Reading graphs for limits:

Follow the curve as you approach from each side — ignore any "dots" at the point
Open circle: Function undefined there, but limit may exist
Jump in graph: Left and right limits differ → limit DNE
Vertical line (asymptote): Limit approaches ±∞

Trick 3: Quick substitution test:

For continuous functions (polynomials, trig functions at valid points), \(\lim_{x \to a} f(x) = f(a)\)
If you get \(\frac{0}{0}\) (indeterminate form), factor and simplify first
If you get \(\frac{\text{nonzero}}{0}\), the limit is ±∞ (vertical asymptote)

❌ Common Mistakes to Avoid

Mistake 1: Confusing \(\lim_{x \to a} f(x)\) with \(f(a)\) — they are NOT the same!
Mistake 2: Writing \(x = a\) instead of \(x \to a\) in limit notation
Mistake 3: Forgetting to check BOTH one-sided limits for piecewise functions
Mistake 4: Saying limit DNE when it equals \(\infty\) — be specific: "approaches infinity"
Mistake 5: Plugging in \(x = a\) directly when you get \(\frac{0}{0}\) — must simplify first!
Mistake 6: Confusing \(x \to a^-\) (left) with \(x \to a^+\) (right)

🔀 Does the Limit Exist? Decision Guide

Follow this process to determine if a limit exists:

Find the left-hand limit: \(\lim_{x \to a^-} f(x) = L_1\)
Find the right-hand limit: \(\lim_{x \to a^+} f(x) = L_2\)
Compare:
- If \(L_1 = L_2\) (both finite and equal) → Limit EXISTS and equals that common value
- If \(L_1 \neq L_2\) → Limit DNE (jump discontinuity)
- If either is ±∞ → Limit DNE (or "approaches infinity")

🌟 Special Notations to Know

Situation	Notation	Interpretation
Direct Substitution Works	\(\lim_{x \to a} f(x) = f(a)\)	Function is continuous at \(a\)
Indeterminate Form	\(\frac{0}{0}\) or \(\frac{\infty}{\infty}\)	Must use algebra to simplify
Removable Discontinuity	\(\lim_{x \to a} f(x) = L\), \(f(a)\) undefined	Hole in graph at \((a, L)\)
Jump Discontinuity	\(\lim_{x \to a^-} \neq \lim_{x \to a^+}\)	Limit DNE; graph "jumps"
Vertical Asymptote	\(\lim_{x \to a} f(x) = \pm\infty\)	Function unbounded near \(x = a\)
Horizontal Asymptote	\(\lim_{x \to \pm\infty} f(x) = L\)	Function approaches \(y = L\) at ends

📖 Practice Reading Limit Statements

Statement: \(\lim_{x \to 5^-} f(x) = 3\)

Read as: "As \(x\) approaches 5 from the left, \(f(x)\) approaches 3"

Meaning: When \(x\) values slightly less than 5 (like 4.9, 4.99) are substituted, \(f(x)\) gets close to 3

Statement: \(\lim_{x \to \infty} \frac{1}{x} = 0\)

Read as: "As \(x\) approaches infinity, \(\frac{1}{x}\) approaches 0"

Meaning: As \(x\) gets very large, \(\frac{1}{x}\) gets closer and closer to 0 (horizontal asymptote at \(y = 0\))

Statement: \(\lim_{x \to 2} \frac{1}{x-2} = \infty\)

Read as: "As \(x\) approaches 2, \(\frac{1}{x-2}\) approaches infinity"

Meaning: There is a vertical asymptote at \(x = 2\); the function grows without bound

⚡ Quick Reference Card

You Want To...	Write This	Remember
Find a regular limit	\(\lim_{x \to a} f(x) = L\)	Check both sides match
Find left-hand limit	\(\lim_{x \to a^-} f(x) = L\)	Approach from \(x < a\)
Find right-hand limit	\(\lim_{x \to a^+} f(x) = L\)	Approach from \(x > a\)
Say limit doesn't exist	\(\lim_{x \to a} f(x)\) DNE	Left ≠ Right, or oscillation
Describe vertical asymptote	\(\lim_{x \to a} f(x) = \pm\infty\)	Unbounded behavior
Describe horizontal asymptote	\(\lim_{x \to \pm\infty} f(x) = L\)	End behavior of function

🔗 Connection to Future Topics

Unit 1.2 sets the foundation for:

Unit 1.3: Estimating limits from graphs and tables
Unit 1.4: Limit properties and algebraic manipulation
Unit 1.5-1.16: Continuity, discontinuities, and advanced limit theorems
Unit 2: The derivative (uses the limit definition)
Understanding notation is critical for all of calculus!

Remember: Limit notation is the language of calculus. Understanding how to read, write, and interpret limits is essential for success in AP® Calculus AB and BC. Master this notation now, and everything else will follow! 🎯