unit-1 – HighFiveAP

1.1 - 1.4

The Definition of a Limit

Calculus is built on the concept of the limit. Unlike Algebra, which deals with static values, Calculus studies dynamic trends. When you see the lim notation, think "approaching," not "arriving."

Formal Definition

We write lim_x→c f(x) = L if and only if the value of f(x) gets arbitrarily close to L as x gets sufficiently close to c from both sides.

lim_x→c f(x) = L ⟺ lim_x→c^- f(x) = lim_x→c⁺ f(x) = L

⚠️ Crucial Distinctions:

The Limit (L): Where the function is heading.
The Function Value (f(c)): Where the function actually is.
These two do NOT have to be the same! A limit can exist even if there is a hole at x=c.

1.5 - 1.9

Calculating Limits Algebraically

When you plug in x = c and get 0/0, this is called an Indeterminate Form. It does not mean "undefined" or "does not exist." It means "do more work" to remove the zero factor.

Strategy: The "Big Three" Algebraic Fixes

Method	When to Use	Example Strategy
1. Factoring	Polynomials	(x²-9)/(x-3) → (x-3)(x+3)/(x-3). Cancel the (x-3) term.
2. Conjugates	Square Roots	Multiply numerator and denominator by √(x+a) + b to clear the root.
3. Trig Identities	Trig Functions	Use special limits: lim_x→0 (sin x)/x = 1 or lim_x→0 (1-cos x)/x = 0.

The Squeeze Theorem: Used for oscillating functions like x² sin(1/x) as x→0. Since -x² ≤ x² sin(1/x) ≤ x², and both ends go to 0, the middle function must also go to 0.

1.10 - 1.13

Continuity

This is the most frequent FRQ topic in Unit 1. You cannot simply say "the graph is unbroken." You must use the formal 3-step definition to prove it.

★ The Continuity Checklist (Memorize This!)

f(c) is Defined
The point exists.
(No hole, no asymptote)

Limit Exists
lim_x→c⁻ = lim_x→c⁺
Left meets Right.

Limit = Value
lim_x→c f(x) = f(c)
The bridge connects.

Types of Discontinuities

[Image of types of discontinuities calculus]

Removable (Hole): The limit exists, but f(c) is missing or wrong. (Common in rational functions).
Jump: Left limit ≠ Right limit. (Common in piecewise functions and |x|/x).
Infinite (Asymptote): At least one side goes to ∞ or -∞.

1.14 - 1.15

Asymptotes & Infinity

In Calculus, asymptotes are strictly defined using limits.

Type	Calculus Definition	Rational Function Shortcut
Vertical Asymptote (Infinite Limit)	lim_x→c f(x) = ±∞	Denominator is 0 (and numerator is non-zero). *Always factor first!
Horizontal Asymptote (Limit at Infinity)	lim_x→∞ f(x) = L	Compare Degrees: • Top > Bot: ±∞ (No HA) • Bot > Top: 0 • Equal: Ratio of Coefficients

1.16

Intermediate Value Theorem (IVT)

The IVT is an Existence Theorem. It guarantees that a value exists, but does not calculate it.

Official Statement

If f(x) is continuous on the closed interval [a, b], and k is any number between f(a) and f(b), then there exists at least one number c in [a, b] such that f(c) = k.

FRQ Justification Template:
Problem: Prove there is a c in [2, 5] where f(c) = 7.
1. Condition: "Since f is continuous on [2, 5]..." (Must be stated!)
2. Values: "We find that f(2) = 4 and f(5) = 10."
3. Conclusion: "Since 4 < 7 < 10, by the Intermediate Value Theorem, there exists a c such that f(c)=7."