Pre-Calculus - Week 12

Limits Preview and Calculus Readiness

This final week is the bridge to Calculus I. You have spent eleven weeks building the algebra, graphing, and trigonometry that calculus requires. Now you get an honest preview of limits — the definition, how to evaluate them numerically and algebraically, and the intuition of what happens when x approaches a value without reaching it. Students who understand this week enter Calculus I with a significant advantage.

Learning Objectives

- Understand the informal definition of a limit and what it means for x to approach a value. - Evaluate limits numerically using tables of values. - Evaluate limits graphically. - Evaluate limits algebraically by direct substitution. - Recognize and resolve indeterminate forms by factoring. - Understand the concept of continuity and how limits relate to it. - Estimate instantaneous rate of change as the limit of average rates of change. - Summarize the prerequisite algebra and trig skills for Calculus I.

1. The Idea of a Limit

A limit asks: what value does f(x) approach as x gets close to some number a? We write: lim x->a f(x) = L. Important: we are NOT asking what f(a) equals. We ask what the OUTPUT approaches as the INPUT gets close to a. Example: f(x) = x + 3. As x approaches 2, f(x) approaches 5. lim x->2 (x + 3) = 5. This is obvious for polynomial functions. Limits become interesting when f(a) is undefined or when there is a hole or jump in the graph.

2. Evaluating Limits Numerically

Build a table of values with x approaching a from both sides. Example: Estimate lim x->2 (x^2 - 4)/(x - 2). x values: 1.9, 1.99, 1.999, 2.001, 2.01, 2.1 f(x) values: 3.9, 3.99, 3.999, 4.001, 4.01, 4.1 The outputs approach 4 from both sides. So lim x->2 (x^2-4)/(x-2) = 4. Note: at x=2, the function is 0/0 (undefined). But the LIMIT still exists because the function approaches a value.

3. One-Sided Limits

Left-hand limit: lim x->a^(-) f(x) — approaching a from the left (values less than a). Right-hand limit: lim x->a^(+) f(x) — approaching a from the right (values greater than a). The two-sided limit lim x->a f(x) exists if and only if both one-sided limits are equal. Example: f(x) = {x + 1 if x < 2; x^2 - 1 if x >= 2}. Left: lim x->2^(-) = 2 + 1 = 3. Right: lim x->2^(+) = 4 - 1 = 3. Two-sided limit exists: lim x->2 f(x) = 3. But f(2) = 4 - 1 = 3 as well. So the function IS continuous at x=2.

4. Direct Substitution

For polynomial and rational functions (where the denominator is not zero), simply substitute. Example 1: lim x->3 (x^2 + 2x - 1). Substitute: 9 + 6 - 1 = 14. Example 2: lim x->1 (x^3 - x)/(x + 1). Substitute: (1 - 1)/2 = 0/2 = 0. Example 3: lim x->-2 (x^2 + 3x + 2)/(x + 2). Substitute: (4 - 6 + 2)/0 = 0/0. Indeterminate form. Need more work.

5. Resolving 0/0 by Factoring

When direct substitution gives 0/0, factor and cancel. Example 1: lim x->2 (x^2 - 4)/(x - 2). Factor: (x-2)(x+2)/(x-2) = x+2 (for x not equal 2). Substitute: 2 + 2 = 4. Example 2: lim x->-2 (x^2 + 3x + 2)/(x + 2). Factor numerator: (x+2)(x+1)/(x+2) = x+1. Substitute: -2 + 1 = -1. Example 3: lim x->1 (x^3 - 1)/(x - 1). Factor: (x-1)(x^2+x+1)/(x-1) = x^2+x+1. Substitute: 1 + 1 + 1 = 3.

6. Limits at Infinity

lim x->infinity f(x) describes the end behavior of f(x). For rational functions: - If degree of numerator < denominator: limit is 0. - If degrees are equal: limit is ratio of leading coefficients. - If degree of numerator > denominator: limit is ±infinity (no finite limit). Example 1: lim x->inf (3x^2)/(x^2 + 1). Divide numerator and denominator by x^2: 3/(1 + 1/x^2) -> 3/1 = 3. Example 2: lim x->inf (5x + 1)/(x^2 - 3). Divide by x^2: (5/x + 1/x^2)/(1 - 3/x^2) -> 0/1 = 0.

7. Continuity

A function f is continuous at x = a if three conditions hold: 1. f(a) is defined. 2. lim x->a f(x) exists. 3. lim x->a f(x) = f(a). If any condition fails, f has a discontinuity at a. Types of discontinuity: - Removable (hole): lim exists but f(a) is undefined or different. You can 'remove' it by redefining f(a). - Jump: left and right limits exist but are not equal. - Infinite: the limit is ±infinity (vertical asymptote). Polynomials are continuous everywhere. Rational functions are continuous on their domain.

8. Instantaneous Rate of Change

Average rate of change of f on [a, a+h]: ARC = (f(a+h) - f(a))/h. As h approaches 0, this becomes the instantaneous rate of change — the derivative. Example: f(x) = x^2. Find the average rate of change from x=2 to x=2+h. f(2) = 4, f(2+h) = (2+h)^2 = 4 + 4h + h^2. ARC = (4 + 4h + h^2 - 4)/h = (4h + h^2)/h = 4 + h. As h->0, ARC approaches 4. So the instantaneous rate of change at x=2 is 4. This is exactly what calculus does: lim h->0 (f(a+h)-f(a))/h = f'(a).

9. Algebra Review for Calculus

Calculus I assumes mastery of these Pre-Calculus skills: 1. Factoring: trinomials, difference of squares, sum/difference of cubes. 2. Simplifying rational expressions. 3. Exponent rules (especially negative and fractional exponents). 4. Solving equations (linear, quadratic, rational, exponential, logarithmic). 5. Function composition and inverses. 6. Graphing: lines, parabolas, exponentials, logs. 7. Trig values for 30°, 45°, 60° and their quadrant analogs. 8. Pythagorean and reciprocal identities. Weak algebra is the #1 reason students struggle in Calculus I — not the calculus itself.

10. Trig Review for Calculus

Calculus I uses trig in: - Derivatives of sin(x) and cos(x) (you will need to know d/dx sin(x) = cos(x) and d/dx cos(x) = -sin(x)). - Integration: integrals of trig functions. - Related rates problems with angles. - Trig substitution in integration (Calculus II). Before starting Calculus I, you should be able to: - State sin, cos, tan for 0°, 30°, 45°, 60°, 90°, and their quadrant analogs, from memory. - Apply Pythagorean identities without hesitation. - Graph y = sin(x) and y = cos(x) and state their key features.

11. What's Next: Calculus I

Calculus I topics in order: 1. Limits (formal definition, properties, L'Hôpital's rule). 2. Derivatives (limit definition, power rule, product/quotient/chain rules). 3. Applications of derivatives (optimization, related rates, curve sketching). 4. Antiderivatives and integration. 5. The Fundamental Theorem of Calculus (connecting derivatives and integrals). The most common student mistake: underestimating how much algebra and trig you need during calculus. The calculus concepts are not that hard. The algebra errors on every step cost points. You are now ready. The foundation is solid.