Fundamental Limit Laws

Why This Matters

Limit laws are the computational engine of calculus—they're how you actually evaluate limits without relying on tables or graphs. Every derivative you'll compute later in the course depends on these rules working seamlessly in the background. When you're tested on limits, you're really being tested on whether you can break complex expressions into manageable pieces and recognize when standard rules apply (and when they don't).

These laws demonstrate a fundamental principle: limits respect algebraic operations. That means you can pull limits apart, work with the pieces, and reassemble them—as long as certain conditions are met. The exam loves to test those conditions, especially the restrictions on the quotient rule and composite functions. Don't just memorize formulas; know why each rule works and when it fails.

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Building Block Laws

These are the foundation—simple rules that establish how limits behave for the most basic functions. Every other limit law builds on these.

Limit of a Constant

• The limit of any constant is itself—no matter what value $x$ approaches, $\lim_{x \to c} k = k$
• Constants don't "move" as $x$ changes, so there's nothing to approach; the function is already at $k$
• Foundation for the constant multiple rule—this explains why you can factor constants out of limits later

Limit of the Identity Function

• The limit of $x$ as $x \to c$ equals $c$—written as $\lim_{x \to c} x = c$
• The identity function $f(x) = x$ is continuous everywhere, so the limit equals the function value
• Basis for polynomial limits—combined with power and sum rules, this lets you evaluate any polynomial by direct substitution

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Arithmetic Operation Laws

These rules let you break apart complex limit expressions into simpler pieces. The key insight: limits distribute over basic operations, turning one hard problem into several easy ones.

Sum/Difference Rule

• Limits split across addition and subtraction—$\lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)$
• Both individual limits must exist for this rule to apply; if either is undefined, you need a different approach
• Essential for polynomials—lets you evaluate $\lim_{x \to 2}(x^2 + 3x - 5)$ term by term

Product Rule

• Limits distribute over multiplication—$\lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$
• Both limits must exist and be finite—watch for indeterminate forms like $0 \cdot \infty$
• Works for any finite number of factors—extend it to three or more functions as needed

Quotient Rule

• Limits distribute over division—$\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$
• Critical restriction: denominator limit cannot be zero—if $\lim_{x \to c} g(x) = 0$, this rule doesn't apply
• When both limits are zero, you have the indeterminate form $\frac{0}{0}$, requiring algebraic manipulation or L'Hôpital's Rule

Constant Multiple Rule

• Constants factor out of limits—$\lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x)$
• Direct consequence of the product rule combined with the limit of a constant
• Simplifies computation—pull out coefficients first, then focus on the variable expression

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Power and Root Laws

These rules handle exponents and radicals, extending your toolkit to non-linear expressions. The principle: if you can find the limit of the base, you can find the limit of any power or root of it.

Power Rule

• Exponents pass through limits—$\lim_{x \to c} [f(x)]^n = \left[\lim_{x \to c} f(x)\right]^n$ for positive integer $n$
• The base limit must exist—if $\lim_{x \to c} f(x)$ is undefined, you can't apply this rule
• Extends to polynomials—combined with sum and constant rules, this is why $\lim_{x \to c} p(x) = p(c)$ for any polynomial

Root Rule

• Roots pass through limits—$\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to c} f(x)}$
• Restriction for even roots: the limit of the radicand must be non-negative (you can't take $\sqrt{-4}$ in real numbers)
• Odd roots are more flexible—$\sqrt[3]{-8} = -2$ is valid, so odd root rules apply to negative limits too

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Advanced Limit Techniques

These rules handle situations where basic arithmetic laws aren't enough—nested functions and functions that resist direct evaluation.

Limits of Composite Functions

• For continuous outer functions, $\lim_{x \to c} f(g(x)) = f\left(\lim_{x \to c} g(x)\right)$
• Continuity of $f$ at the inner limit is required—if $\lim_{x \to c} g(x) = L$, then $f$ must be continuous at $L$
• Powerful for nested expressions—evaluate $\lim_{x \to 0} \sin(x^2)$ by first finding $\lim_{x \to 0} x^2 = 0$, then $\sin(0) = 0$

Squeeze Theorem

• Traps a function between two others—if $h(x) \leq f(x) \leq g(x)$ and $\lim_{x \to c} h(x) = \lim_{x \to c} g(x) = L$, then $\lim_{x \to c} f(x) = L$
• Essential for oscillating functions—classic example: $\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right) = 0$ because $-x^2 \leq x^2\sin\left(\frac{1}{x}\right) \leq x^2$
• Both bounding limits must equal the same value—if they differ, the squeeze doesn't work

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Quick Reference Table

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Self-Check Questions

1. Which two limit laws together explain why you can evaluate $\lim_{x \to 3} 5x^2$ by computing $5 \cdot 9 = 45$?

2. The Quotient Rule requires a specific condition on the denominator. What is it, and what happens when that condition fails with the numerator also approaching zero?

3. Compare and contrast the Power Rule and Root Rule—what additional restriction applies to even roots that doesn't apply to powers?

4. You need to evaluate $\lim_{x \to 0} x \sin\left(\frac{1}{x}\right)$. Which limit law applies here, and what bounding functions would you use?

5. If $g(x)$ is continuous at $x = 2$ and $\lim_{x \to 2} f(x) = 5$, under what condition can you conclude that $\lim_{x \to 2} g(f(x)) = g(5)$?