Unit 01: Limits & Continuity

Question 1

Limit: General Definition and Notation

Answer 1

Definition: The value (L) a function approaches as x gets closer to a specific number (c) from both sides. Notation: The limit as x approaches c of f(x) equals L. Note: The function does not have to exist at ‘c’ for the limit to exist.

Question 2

Three Reasons a Limit Does Not Exist (DNE)

Answer 2

1. Different Left and Right Behavior: The function approaches different values from each side. 2. Oscillating Behavior: The function fluctuates too rapidly. 3. Unbounded Behavior: The function goes to positive or negative infinity.

Question 3

Strategy: Direct Substitution

Answer 3

The first step in evaluating any limit. Plug the target x-value directly into the function. If you get a real number, that is your limit. If you get 0/0, you have an Indeterminate Form.

Question 4

Indeterminate Form (0/0) and Next Steps

Answer 4

Definition: A result that provides no information about the limit. AP Application: You must perform further algebraic work, specifically the Cancellation Technique (factoring) or the Rationalization Technique (conjugates).

Question 5

Cancellation Technique (Factoring)

Answer 5

An algebraic method used when direct substitution gives 0/0. Factor the numerator and denominator to ‘cancel’ the term causing the zero, then re-apply direct substitution.

Question 6

Rationalization Technique (Conjugates)

Answer 6

Used when a limit involves square roots and yields 0/0. Multiply the numerator and denominator by the conjugate (the same expression but with the opposite middle sign) to eliminate the radical.

Question 7

Existence of a Limit: The L=R Rule

Answer 7

Requirement: A general limit exists if and only if the Left-Sided Limit equals the Right-Sided Limit. If the two sides approach different values, the general limit is DNE.

Question 8

Three Criteria for Continuity at a Point

Answer 8

A function is continuous at point ‘c’ if: 1. f(c) is defined (no hole). 2. The limit as x approaches c exists (sides meet). 3. The limit equals the function value (the hole is filled).

Question 9

Intermediate Value Theorem (IVT)

Answer 9

Definition: If f is continuous on [a, b], then the function must hit every y-value between f(a) and f(b) at least once. AP Application: Often used to prove a ‘root’ (zero) exists if f(a) is negative and f(b) is positive.

Question 10

Vertical Asymptote vs. Removable Discontinuity (Hole)

Answer 10

Hole: Occurs when a factor in the denominator cancels out with the numerator (0/0). Vertical Asymptote: Occurs when a factor in the denominator remains and causes division by zero (n/0).

Question 11

Infinite Limits: Definition and Meaning

Answer 11

Describes behavior at a vertical asymptote where the function increases/decreases without bound. Note: While we write the limit as ∞ or -∞, the limit technically ‘Does Not Exist’ because infinity is not a finite number.

Question 12

Limits at Infinity (End Behavior)

Answer 12

Describes the value a function approaches as x moves toward ±∞. This value represents the Horizontal Asymptote of the graph.

Question 13

Horizontal Asymptote Rule: Bottom Heavy

Answer 13

If the degree of the denominator is higher than the numerator, the limit as x approaches infinity is 0 (y = 0).

Question 14

Horizontal Asymptote Rule: Balanced

Answer 14

If the degrees of the numerator and denominator are equal, the limit as x approaches infinity is the ratio of the leading coefficients.

Question 15

Horizontal Asymptote Rule: Top Heavy

Answer 15

If the degree of the numerator is higher than the denominator, the limit as x approaches infinity is ±∞ (DNE), and there is no horizontal asymptote.

Question 16

Continuity on a Closed Interval [a, b]

Answer 16

A function is continuous on [a, b] if it is continuous at every interior point, and the one-sided limits at the endpoints (a and b) match the function values f(a) and f(b).