Limits Rules

Why This Matters

Limits are the gateway to calculus—they're the foundational concept that makes derivatives, integrals, and continuity possible. In Honors Pre-Calc, you're being tested on your ability to evaluate limits using specific rules, recognize when those rules apply, and understand why functions behave the way they do as they approach certain values. This isn't just about plugging numbers into formulas; it's about developing the analytical thinking that will carry you through AP Calculus.

The rules you'll learn here fall into distinct categories: basic building blocks, algebraic operations, special techniques, and asymptotic behavior. Each rule exists because of how functions fundamentally behave near specific points or at extreme values. Don't just memorize the formulas—know which rule to reach for in different situations and understand what each one tells you about the function's behavior. That's what separates students who struggle with limits from those who master them.

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Basic Building Blocks

These foundational rules establish how the simplest functions behave as $x$ approaches a value. Every complex limit evaluation builds on these two principles.

Limit of a Constant

• The limit equals the constant itself—no matter what value $x$ approaches, a constant function never changes
• Formal notation: $\lim_{x \to c} k = k$, where $k$ is any constant
• Why it matters: This rule lets you pull constants out of more complex limit expressions

Limit of x

• The limit of $x$ as $x \to c$ is simply $c$—the function value equals the input at that point
• Formal notation: $\lim_{x \to c} x = c$, reflecting direct substitution for continuous functions
• Foundation for polynomials: Combined with the power rule, this lets you evaluate any polynomial limit directly

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Algebraic Operation Rules

These rules let you break apart complex expressions into simpler pieces. The key insight: limits distribute across basic arithmetic operations, allowing you to evaluate piece by piece.

Constant Multiple Rule

• Pull constants outside the limit—multiply after evaluating the function's limit
• Formal notation: $\lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x)$
• Exam strategy: Always simplify by factoring out constants first to reduce calculation errors

Sum and Difference Rule

• Limits split across addition and subtraction—evaluate each piece separately, then combine
• Formal notation: $\lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)$
• Application: This is how you handle polynomial limits—break them into individual terms

Product Rule

• Multiply the individual limits—each function's limit is evaluated independently
• Formal notation: $\lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$
• Critical condition: Both individual limits must exist for this rule to apply

Quotient Rule

• Divide the individual limits—but only when the denominator's limit isn't zero
• Formal notation: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$, provided $\lim_{x \to c} g(x) \neq 0$
• Watch out: When the denominator approaches zero, you may have an indeterminate form or infinite limit instead

Power Rule

• Raise the limit to the power—exponents pass through the limit operation
• Formal notation: $\lim_{x \to c} [f(x)]^n = \left[\lim_{x \to c} f(x)\right]^n$
• Polynomials made easy: Combined with the limit of $x$, this handles any term like $x^3$ or $x^{10}$

Root Rule

• Take the root of the limit—radicals pass through when the limit exists and is valid
• Formal notation: $\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to c} f(x)}$
• Domain restriction: For even roots, the limit inside must be non-negative

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Special Techniques

When direct substitution fails or functions behave unusually, these tools provide alternative approaches. These are the problem-solving strategies that separate routine problems from challenging ones.

One-Sided Limits

• Approach from one direction only—left-hand ($x \to c^-$) or right-hand ($x \to c^+$)
• Notation matters: $\lim_{x \to c^-} f(x)$ means approaching from values less than $c$
• Key test for existence: A two-sided limit exists only if both one-sided limits exist and are equal

Squeeze Theorem

• Trap a function between two others—if the outer functions share a limit, the middle function does too
• Formal statement: If $g(x) \leq f(x) \leq h(x)$ and $\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$, then $\lim_{x \to c} f(x) = L$
• Classic application: Proving $\lim_{x \to 0} x \sin\left(\frac{1}{x}\right) = 0$ by bounding with $\pm|x|$

Limits Involving Trigonometric Functions

• Memorize the key result: $\lim_{x \to 0} \frac{\sin x}{x} = 1$ (with $x$ in radians)
• Related identity: $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$ appears frequently in derivative proofs
• Why it matters: These limits form the foundation for differentiating trig functions in calculus

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Indeterminate Forms and Resolution

When direct substitution gives you something meaningless like $\frac{0}{0}$, these concepts help you find the actual limit. Recognizing indeterminate forms is the first step; resolving them is where the real work begins.

Indeterminate Forms

• Common types: $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \cdot \infty$, $\infty - \infty$, $0^0$, $1^\infty$, $\infty^0$
• Not an answer: These forms mean the limit requires additional work—factoring, conjugates, or L'Hôpital's rule
• Detection method: Always try direct substitution first to identify if you have an indeterminate form

L'Hôpital's Rule

• Take derivatives of top and bottom separately—only for $\frac{0}{0}$ or $\frac{\infty}{\infty}$ forms
• Formal statement: If $\lim_{x \to c} \frac{f(x)}{g(x)}$ yields $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$
• Can repeat: If the result is still indeterminate, apply the rule again until you get a determinate form

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Asymptotic Behavior

These concepts describe what happens at the extremes—as $x$ grows without bound or as functions blow up near certain points. Asymptotes are the graphical representation of these limit behaviors.

Limits at Infinity

• End behavior analysis: $\lim_{x \to \infty} f(x)$ and $\lim_{x \to -\infty} f(x)$ describe what happens as $x$ grows large
• Horizontal asymptotes: If $\lim_{x \to \pm\infty} f(x) = L$, then $y = L$ is a horizontal asymptote
• Rational function shortcut: Compare the degrees of numerator and denominator to quickly find these limits

Infinite Limits

• Function values blow up: $\lim_{x \to c} f(x) = \infty$ or $-\infty$ means unbounded growth near $x = c$
• Vertical asymptotes: These limits indicate a vertical asymptote at $x = c$
• One-sided behavior: Often $\lim_{x \to c^+} f(x) = \infty$ while $\lim_{x \to c^-} f(x) = -\infty$ (or vice versa)

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Quick Reference Table

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Self-Check Questions

1. Which two rules both require you to evaluate individual limits before combining them, but one has a critical restriction the other doesn't? What is that restriction?

2. You evaluate $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ by direct substitution and get $\frac{0}{0}$. Is this the final answer? What should you do next, and what concept does this illustrate?

3. Compare and contrast limits at infinity and infinite limits. If a function has a horizontal asymptote at $y = 3$ and a vertical asymptote at $x = -1$, write the limit notation that describes each.

4. A limit problem gives you $\lim_{x \to 0^+} f(x) = 4$ and $\lim_{x \to 0^-} f(x) = -2$. Does $\lim_{x \to 0} f(x)$ exist? Which rule or concept justifies your answer?

5. You need to find $\lim_{x \to 0} x^2 \cos\left(\frac{1}{x}\right)$. Direct substitution doesn't work, and you can't factor. Which technique from this guide would you use, and what two bounding functions would help you?