Key Concepts of Limits of Functions

Why This Matters

Limits are the gateway to calculus—they're the mathematical tool that lets us analyze what happens at boundaries, near holes in graphs, and as values stretch toward infinity. In Precalculus, you're being tested on your ability to evaluate limits using multiple methods, identify when limits exist or fail to exist, and connect limit behavior to continuity and asymptotes. These concepts show up repeatedly in AP Calculus, so mastering them now gives you a serious head start.

The key insight is that limits describe behavior near a point, not necessarily at the point itself. This distinction trips up many students. As you work through these concepts, don't just memorize formulas—understand why a limit might not exist, how different evaluation techniques connect, and what the relationship between limits and continuity really means. That conceptual understanding is what separates students who ace limit problems from those who struggle.

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Foundational Definitions

Before you can evaluate limits, you need rock-solid understanding of what they actually mean. A limit captures the trend of a function's output as the input approaches—but doesn't necessarily reach—a specific value.

Definition of a Limit

• The formal notation $\lim_{x \to c} f(x) = L$ means that as $x$ gets arbitrarily close to $c$, the output $f(x)$ gets arbitrarily close to $L$
• Limits can be finite or infinite—a finite limit means the function settles toward a real number; an infinite limit means the function grows without bound
• The limit value $L$ may differ from $f(c)$—this is crucial for understanding discontinuities and why limits matter in the first place

One-Sided Limits

• Left-hand limit $\lim_{x \to c^-} f(x)$ examines behavior as $x$ approaches $c$ from values less than $c$; right-hand limit $\lim_{x \to c^+} f(x)$ approaches from values greater than $c$
• A two-sided limit exists only when both one-sided limits exist and are equal—this is a common exam question setup
• Jump discontinuities occur when one-sided limits exist but differ—the function "jumps" from one value to another at that point

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Computational Tools

These are your workhorses for actually calculating limits. Limit laws let you break complex expressions into manageable pieces, while algebraic techniques handle the tricky cases where direct substitution fails.

Limit Laws (Sum, Difference, Product, Quotient)

• Sum/Difference Law: $\lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)$—you can evaluate limits of each piece separately
• Product Law: $\lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$—multiplication distributes across limits
• Quotient Law: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$ only when the denominator's limit is nonzero—this restriction is heavily tested

Evaluating Limits Algebraically

• Direct substitution is your first move—if plugging in $c$ gives a real number, that's your limit; no further work needed
• Indeterminate forms like $\frac{0}{0}$ signal that algebraic manipulation is required—try factoring, expanding, or rationalizing
• Rationalizing (multiplying by the conjugate) is essential for limits involving square roots—this technique eliminates problematic radicals

Evaluating Limits Graphically

• Trace the y-values as x approaches the target—the limit is what the function approaches, even if there's a hole at that point
• Vertical asymptotes indicate infinite limits—the function blows up toward $\pm\infty$ rather than approaching a finite value
• Holes vs. asymptotes look different graphically—holes are single missing points; asymptotes show the function shooting off toward infinity

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Special Techniques

Some limits resist standard approaches and require specialized theorems. These techniques handle oscillating functions, bounded expressions, and the fundamental trigonometric limits that appear constantly in calculus.

Squeeze Theorem

• The setup: if $g(x) \leq f(x) \leq h(x)$ near $c$, and $\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$, then $\lim_{x \to c} f(x) = L$
• Use this for oscillating functions—especially expressions like $x \sin\left(\frac{1}{x}\right)$ where direct evaluation is impossible
• You must establish the bounding inequalities first—the theorem only works when you can prove the function is trapped between two others

Limits of Trigonometric Functions

• Memorize this: $\lim_{x \to 0} \frac{\sin x}{x} = 1$—this fundamental limit appears everywhere in calculus and is frequently tested
• Second key limit: $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$ and $\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$—know both versions
• Manipulate expressions to match these forms—factor out constants, substitute variables, or rewrite using trig identities

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Behavior at Infinity

Limits at infinity reveal what happens to functions in the long run—their end behavior. This connects directly to horizontal asymptotes and helps you understand the overall shape of function graphs.

Limits Involving Infinity

• $\lim_{x \to \infty} f(x) = L$ means the function levels off—as $x$ grows without bound, $f(x)$ approaches the finite value $L$
• Horizontal asymptotes occur when this limit exists—the line $y = L$ represents the function's long-term behavior
• $\lim_{x \to c} f(x) = \infty$ describes vertical asymptotic behavior—the function grows without bound as $x$ approaches $c$

Limits at Infinity for Polynomials and Rationals

• For polynomials, the leading term dominates—$\lim_{x \to \infty} (3x^4 - 2x + 1)$ behaves like $\lim_{x \to \infty} 3x^4 = \infty$
• For rational functions, compare degrees—if numerator degree < denominator degree, limit is 0; if equal, limit is ratio of leading coefficients; if numerator degree > denominator degree, limit is $\pm\infty$
• Divide by the highest power of $x$ in the denominator—this technique transforms the expression into something you can evaluate directly

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Connecting Limits to Continuity

The relationship between limits and continuity is one of the most important conceptual links in precalculus. Continuity is defined through limits, and understanding discontinuities requires understanding where limits fail.

Continuity and Limits

• Three conditions for continuity at $x = c$: (1) $f(c)$ is defined, (2) $\lim_{x \to c} f(x)$ exists, (3) $\lim_{x \to c} f(x) = f(c)$
• Removable discontinuities occur when the limit exists but doesn't equal the function value—you could "fill in" the hole
• Jump discontinuities occur when one-sided limits exist but differ; infinite discontinuities occur at vertical asymptotes where limits don't exist

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Quick Reference Table

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Self-Check Questions

1. What three conditions must be satisfied for a function to be continuous at a point, and which condition typically fails for a removable discontinuity?

2. If $\lim_{x \to 3^-} f(x) = 5$ and $\lim_{x \to 3^+} f(x) = 7$, does $\lim_{x \to 3} f(x)$ exist? What type of discontinuity does this describe?

3. Compare and contrast the Squeeze theorem and the fundamental trigonometric limit $\lim_{x \to 0} \frac{\sin x}{x} = 1$—when would you use each?

4. For the rational function $\frac{2x^3 + 1}{5x^3 - x}$, what is the limit as $x \to \infty$, and what does this tell you about the graph?

5. You encounter $\lim_{x \to 4} \frac{x^2 - 16}{x - 4}$ and direct substitution gives $\frac{0}{0}$. Walk through the algebraic technique you'd use and explain why this indeterminate form doesn't mean the limit is undefined.