Unit 1: Limits and Continuity

AP Calculus AB: 80 practice questions with detailed explanations.

Unit Study Guide

Executive Summary

Unit 1 builds the foundation of calculus: limits describe what a function's output approaches as the input approaches a value. Continuity is the bridge from limits to derivatives.

Limits

A limit $\lim_{x \to a} f(x) = L$ means $f(x)$ gets arbitrarily close to $L$ as $x$ approaches $a$ — the value of $f(a)$ itself is irrelevant. One-sided limits ( $x \to a^-$ and $x \to a^+$ ) must agree for the two-sided limit to exist. Limits at ∞ describe end behavior and horizontal asymptotes.

Techniques

Factor and cancel, rationalize, or use a common denominator to resolve $0/0$ forms. The Squeeze Theorem bounds a function between two functions with the same limit.

Continuity

$f$ is continuous at $x = a$ if $f(a)$ exists, the limit exists, and the two agree. Polynomials, rationals (on their domains), trig, exponential, and logarithmic functions are continuous on their domains. Removable (hole), jump, and infinite discontinuities are the three main types.

Big theorems

The Intermediate Value Theorem: if $f$ is continuous on $[a,b]$ and $N$ is between $f(a)$ and $f(b)$ , then $f(c) = N$ for some $c$ in $(a,b)$ .

Exam traps

A limit existing says nothing about $f(a)$ — check the function's value separately. ' $0/0$ ' is an indeterminate form, not a real number. IVT requires continuity on a closed interval — always verify the hypothesis.

Top 5 Concepts to Master

1Evaluate two-sided limits and recognize when they do not exist.
2Resolve 0/0 forms by factoring, rationalizing, or common denominators.
3Classify discontinuities as removable, jump, or infinite.
4Apply the IVT to guarantee a solution exists.
5Use the Squeeze Theorem for oscillating functions.

Key Terms & Definitions

Practice with Flashcards

Limit

Value a function approaches as the input approaches a point.

One-sided limit

Limit as x approaches from the left (x→a⁻) or right (x→a⁺).

Indeterminate form

Expression like 0/0 requiring algebraic manipulation.

Continuity at a point

f(a) defined, limit exists, and they are equal.

Removable discontinuity

A "hole"; the limit exists but the function value differs.

Jump discontinuity

Left and right limits exist but differ.

Infinite discontinuity

Function grows without bound (vertical asymptote).

Intermediate Value Theorem

Continuous f on [a,b] hits every value between f(a) and f(b).

Squeeze Theorem

Bounding function between two functions with a common limit.

Horizontal asymptote

Line y = L approached as x → ±∞.

Vertical asymptote

x = a where the function grows unbounded.

End behavior

Trend of f(x) as x → ∞ or x → −∞.

Common Misconceptions: Exam Traps

If the limit exists, the function is continuous.

Correct: Continuity also requires f(a) to equal the limit.

0/0 means the limit does not exist.

Correct: It is indeterminate — factor or rationalize to resolve it.

IVT applies to any function.

Correct: The function must be continuous on a closed interval.

A function with a hole has a vertical asymptote there.

Correct: Holes are removable; asymptotes are unbounded behavior.

Featured Practice Questions

Question Bank Breakdown

By difficulty

easy 28medium 35hard 17

By topic

Introducing Calculus: Can Change Occur at an Instant? 26Exploring Types of Discontinuities 6Confirming Continuity over an Interval 5Defining Continuity at a Point 5Derivatives of cos(x), sin(x), e^x, and ln(x) 4Defining the Derivative 3Defining Average and Instantaneous Rates of Change 2Connecting Multiple Representations of Limits 2Working with the Intermediate Value Theorem 1Removing Discontinuities 1Introducing Calculus 1Defining Limits and Using Limit Notation 1