AP Precalculus: Limits

Master limit notation, laws, asymptotes, and evaluation techniques

🎯 Definition 📏 Limit Laws 📈 Asymptotes 🔧 Techniques

📚 Understanding Limits

A limit describes the value a function approaches as the input approaches a particular value. Limits are fundamental to calculus and help us understand function behavior near specific points, including where the function may not be defined. They're essential for analyzing continuity, asymptotes, and rates of change.

1 Limit Definition & Notation

The limit of f(x) as x approaches a equals L if f(x) gets arbitrarily close to L as x gets close to a (but not equal to a).

Limit Notation \(\displaystyle\lim_{x \to a} f(x) = L\)

"As x approaches a, f(x) approaches L"

One-Sided Limits

Left-Hand Limit

\(\displaystyle\lim_{x \to a^-} f(x)\)

Approaching a from values less than a

Right-Hand Limit

\(\displaystyle\lim_{x \to a^+} f(x)\)

Approaching a from values greater than a

💡 When Does a Limit Exist?

The limit \(\lim_{x \to a} f(x) = L\) exists if and only if both one-sided limits exist and are equal: \(\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L\)

2 Types of Discontinuities

A discontinuity occurs where a function is not continuous. Understanding discontinuity types helps identify when limits exist or don't exist.

Removable (Hole)

Limit exists but function undefined or has different value at that point

Jump

Left and right limits exist but are different

Infinite (Asymptote)

Function approaches ±∞ as x approaches a

Oscillating

Function oscillates infinitely near the point

📌 Examples

Removable: \(f(x) = \frac{x^2-1}{x-1}\) at x = 1 (hole at (1, 2))

Jump: Piecewise function where pieces don't connect

Infinite: \(f(x) = \frac{1}{x}\) at x = 0

3 Asymptotes & End Behavior

Vertical asymptotes occur where function values approach ±∞. End behavior describes what happens as x → ±∞.

Vertical Asymptote

\(\displaystyle\lim_{x \to a} f(x) = \pm\infty\)

Function unbounded near x = a

Horizontal Asymptote

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

Function approaches L as x → ±∞

End Behavior of Rational Functions

degree(top) < degree(bottom)

Horizontal asymptote at y = 0

degree(top) = degree(bottom)

Horizontal asymptote at y = (leading coefficients ratio)

degree(top) > degree(bottom)

No horizontal asymptote (oblique or no asymptote)

4 Laws of Limits

These properties allow us to break complex limits into simpler parts, provided the individual limits exist and are finite.

Sum Law

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

Difference Law

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

Product Law

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

Quotient Law

\(\lim_{x \to a}\frac{f(x)}{g(x)} = \frac{\lim_{x \to a}f(x)}{\lim_{x \to a}g(x)}\) if denominator ≠ 0

Power Law

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

Root Law

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

Constant Multiple

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

Constant Law

\(\lim_{x \to a} c = c\)

5 Limits of Polynomials & Rational Functions

For polynomials and many rational functions, finding limits is straightforward — just substitute the value. This is called direct substitution.

Polynomial p(x)

\(\displaystyle\lim_{x \to a} p(x) = p(a)\)

Rational Function (if q(a) ≠ 0)

\(\displaystyle\lim_{x \to a} \frac{p(x)}{q(x)} = \frac{p(a)}{q(a)}\)

📌 Example: Direct Substitution

\(\displaystyle\lim_{x \to 3} (x^2 + 2x - 1) = 3^2 + 2(3) - 1 = 9 + 6 - 1 = 14\)

⚠️ Indeterminate Form

If direct substitution gives \(\frac{0}{0}\), you cannot use direct substitution. Use factorization, rationalization, or other techniques to simplify first.

6 Factorization & Rationalization Techniques

When direct substitution gives \(\frac{0}{0}\), algebraic manipulation can often reveal the true limit by canceling common factors.

Method 1: Factorization

Factor numerator and denominator
Cancel the common factor causing 0/0
Substitute to find the limit

📌 Example: Factorization

\(\displaystyle\lim_{x \to 2} \frac{x^2 - 4}{x - 2}\)

Direct sub: \(\frac{0}{0}\) — indeterminate

Factor: \(\frac{(x-2)(x+2)}{x-2} = x + 2\)

Limit: \(\lim_{x \to 2}(x + 2) = 4\)

Method 2: Rationalization

For expressions with square roots, multiply by the conjugate
This eliminates the square root in numerator or denominator
Simplify and then substitute

📌 Example: Rationalization

\(\displaystyle\lim_{x \to 0} \frac{\sqrt{x+4} - 2}{x}\)

Multiply by conjugate: \(\frac{\sqrt{x+4} - 2}{x} \cdot \frac{\sqrt{x+4} + 2}{\sqrt{x+4} + 2}\)

Simplify: \(\frac{(x+4) - 4}{x(\sqrt{x+4} + 2)} = \frac{x}{x(\sqrt{x+4} + 2)} = \frac{1}{\sqrt{x+4} + 2}\)

Limit: \(\lim_{x \to 0}\frac{1}{\sqrt{x+4} + 2} = \frac{1}{2 + 2} = \frac{1}{4}\)

📋 Quick Reference

Limit Notation

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

Left Limit

\(\lim_{x \to a^-} f(x)\)

Right Limit

\(\lim_{x \to a^+} f(x)\)

Limit Exists If

Left limit = Right limit

Polynomial Limit

\(\lim_{x \to a} p(x) = p(a)\)

Indeterminate Form

\(\frac{0}{0}\) → Factor or rationalize