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 in the background.

These laws reflect a fundamental principle: limits respect algebraic operations. You can pull limits apart, work with the pieces, and reassemble them, as long as certain conditions are met. Exams love to test those conditions, especially the restrictions on the quotient rule and composite functions. Don't just memorize formulas; know when each rule applies and when it breaks down.

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Building Block Laws

These are the foundation. Every other limit law builds on these two simple results.

Limit of a Constant

$\lim_{x \to c} k = k$

A constant function doesn't change as $x$ moves, so there's nothing to "approach." The output is always $k$, regardless of what value $x$ is heading toward. This result also underpins the constant multiple rule you'll use constantly.

Limit of the Identity Function

$\lim_{x \to c} x = c$

The identity function $f(x) = x$ is continuous everywhere, so its limit equals its function value. This is the basis for evaluating polynomial limits: combined with the power and sum rules, it's why you can plug in directly.

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Arithmetic Operation Laws

These rules let you break a complex limit expression into simpler pieces. The core idea: limits distribute over basic arithmetic, turning one hard problem into several easy ones.

Sum/Difference Rule

$\lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)$

Both individual limits must exist for this to apply. If either limit is undefined, you need a different approach. This rule is essential for polynomials: it lets you evaluate $\lim_{x \to 2}(x^2 + 3x - 5)$ term by term as $4 + 6 - 5 = 5$.

Product Rule

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

Both limits must exist and be finite. Watch for indeterminate forms like $0 \cdot \infty$, where this rule cannot be applied directly. The rule extends naturally to any finite number of factors.

Quotient Rule

$\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$

Critical restriction: the denominator's limit cannot be zero. If $\lim_{x \to c} g(x) = 0$, this rule doesn't apply. When both numerator and denominator approach zero, you get the indeterminate form $\frac{0}{0}$, which requires algebraic manipulation (factoring, rationalizing) or L'Hôpital's Rule.

Constant Multiple Rule

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

This is actually a special case of the product rule where one "function" is a constant. Recognizing it separately lets you simplify faster: pull out coefficients first, then focus on the variable expression.

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Power and Root Laws

These rules handle exponents and radicals. 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

$\lim_{x \to c} [f(x)]^n = \left[\lim_{x \to c} f(x)\right]^n \quad \text{for positive integer } n$

The base limit must exist. If $\lim_{x \to c} f(x)$ is undefined, you can't apply this rule. Combined with the sum and constant rules, the power rule is why $\lim_{x \to c} p(x) = p(c)$ for any polynomial $p$.

Root Rule

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

For even roots (square root, fourth root, etc.), the limit of the expression under the radical must be non-negative. You can't take $\sqrt{-4}$ in the reals. Odd roots are more flexible since, for example, $\sqrt[3]{-8} = -2$ is perfectly valid.

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Advanced Limit Techniques

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

Limits of Composite Functions

$\lim_{x \to c} f(g(x)) = f\left(\lim_{x \to c} g(x)\right)$

This requires that $f$ is continuous at the value $\lim_{x \to c} g(x) = L$. If $f$ has a discontinuity at $L$, you can't swap the limit inside.

For example, to evaluate $\lim_{x \to 0} \sin(x^2)$:

1. Find the inner limit: $\lim_{x \to 0} x^2 = 0$.
2. Check that $\sin$ is continuous at $0$ (it is).
3. Conclude: $\sin(0) = 0$.

Squeeze Theorem

If $h(x) \leq f(x) \leq g(x)$ near $c$, and $\lim_{x \to c} h(x) = \lim_{x \to c} g(x) = L$, then $\lim_{x \to c} f(x) = L$.

This is your go-to tool for oscillating functions that have no clean algebraic form. The classic example:

$\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right) = 0$

Since $-1 \leq \sin\left(\frac{1}{x}\right) \leq 1$, multiplying through by $x^2$ gives $-x^2 \leq x^2 \sin\left(\frac{1}{x}\right) \leq x^2$. Both $-x^2$ and $x^2$ approach $0$, so the middle expression is squeezed to $0$.

Both bounding limits must equal the same value. If they converge to different numbers, the squeeze doesn't apply.

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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 while the numerator also approaches zero?

3. What additional restriction applies to even roots in the Root Rule that doesn't apply to the Power Rule?

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

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