Limits and Continuity

A High School and Early College Calculus Primer

Limits show up on the very first day of AP Calculus — and they trip up more students than almost any other topic. The notation looks strange, the algebra gets slippery, and most textbooks bury the intuition under pages of formal definitions before you ever see a worked example. This guide cuts straight to what you need.

**TLDR: Limits and Continuity** is a focused, 10–20 page primer written for AP Calculus AB/BC students and Calculus I college students who want to get oriented fast. It covers exactly five things: building real intuition for what a limit is, the algebraic techniques for computing them (substitution, factoring, rationalizing, and the Squeeze Theorem), one-sided and infinite limits with their connection to asymptotes, the three-part definition of continuity and how to classify discontinuities, and the Intermediate Value Theorem with a preview of why all of this matters for derivatives and integrals.

If you are searching for an ap calculus limits and continuity review that respects your time, this is it. Every section leads with the one sentence you need to remember, then unpacks it with worked numbers and concrete examples. Misconceptions are named and corrected directly — no vague warnings, no filler.

This is also a practical calculus 1 limits study guide for students who missed a lecture, are helping a younger sibling, or just want a clean second explanation before an exam.

Pick it up, read it in one sitting, and walk into your next test with the concept locked in.

What you'll learn

Explain what a limit is intuitively and graphically, and why limits exist independent of function values.
Evaluate limits using direct substitution, factoring, rationalizing, and the squeeze theorem.
Distinguish one-sided limits, infinite limits, and limits at infinity, and connect them to vertical and horizontal asymptotes.
Define continuity at a point and on an interval, and classify discontinuities as removable, jump, or infinite.
Apply the Intermediate Value Theorem to guarantee the existence of solutions.

What's inside

1. What a Limit Actually Is

Builds intuition for limits using tables, graphs, and the idea of approaching a value without necessarily reaching it.
2. Computing Limits: Substitution, Algebra, and the Squeeze Theorem

Covers the main techniques for evaluating limits algebraically, including factoring, rationalizing, and bounding.
3. One-Sided Limits, Infinite Limits, and Limits at Infinity

Distinguishes left and right limits, explains vertical asymptotes through infinite limits, and analyzes end behavior with horizontal asymptotes.
4. Continuity at a Point and on an Interval

Defines continuity using the three-part test and classifies the main types of discontinuities.
5. The Intermediate Value Theorem and Why Limits Matter

States and applies the IVT, then previews how limits underpin derivatives and integrals.

Published by Solid State Press

TLDR STUDY GUIDES

Limits and Continuity

A High School and Early College Calculus Primer

Solid State Press

Limits and Continuity

A High School and Early College Calculus Primer

Published by Solid State Press.

Part of the TLDR Study Guides series.

Who This Book Is For

If you're a high school student working through an AP Calculus AB exam prep course, a college freshman looking for a calculus primer that doesn't waste your time, or a tutor who needs a clean review of limits and continuity for a student, this book is for you.

This guide covers everything from how to find limits in calculus using substitution and algebraic techniques, to one-sided limits, limits at infinity and horizontal asymptotes, and the Intermediate Value Theorem explained in plain terms. It also walks through the formal definition of continuity and why it matters for the rest of calculus. Think of it as a focused AP Calculus limits and continuity review — about 15 pages, no padding.

Read it straight through — each section builds on the last. Work every example yourself before reading the solution, then use the problem set at the end to find out what stuck. This calculus study guide for beginners is built for a single focused sitting.

1 What a Limit Actually Is
2 Computing Limits: Substitution, Algebra, and the Squeeze Theorem
3 One-Sided Limits, Infinite Limits, and Limits at Infinity
4 Continuity at a Point and on an Interval
5 The Intermediate Value Theorem and Why Limits Matter

Chapter 1

What a Limit Actually Is

Imagine driving toward a stop sign. You can describe exactly where you're headed — the sign — without ever touching it. That idea, approaching a target value, is the entire engine behind a limit.

Informally: the limit of a function $f(x)$ as $x$ approaches some value $a$ is the number $L$ that $f(x)$ gets arbitrarily close to as $x$ gets closer and closer to $a$. The function does not have to equal $L$ at $x = a$, or even be defined there. The limit is about the journey, not the destination.

The Notation

The standard way to write this is:

$\lim_{x \to a} f(x) = L$

Read it aloud as: "the limit of $f(x)$ as $x$ approaches $a$ equals $L$."

Every piece carries meaning. The $x \to a$ under the limit sign tells you which input value you're approaching. The $L$ on the right is the output value the function is approaching. Neither piece is the function value at $a$ — both are purely about behavior near $a$.

Seeing It in a Table

The most concrete way to check what a limit should be is to plug in values of $x$ that get progressively closer to $a$ and watch what happens to the output.

Consider $f(x) = \dfrac{x^2 - 1}{x - 1}$. At $x = 1$, the denominator is zero, so $f(1)$ is undefined. But that doesn't stop us from asking what $f(x)$ approaches as $x \to 1$.

$x$	$f(x)$
0.9	1.9
0.99	1.99
0.999	1.999
1.001	2.001
1.01	2.01
1.1	2.1

From both sides, $f(x)$ is closing in on $2$. So $\displaystyle\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$, even though $f(1)$ does not exist.

This is worth pausing on. The limit exists and equals $2$. The function value at $x = 1$ does not exist. These are two completely separate questions.

Seeing It on a Graph

On a graph, a limit corresponds to the $y$-value that the curve is visually heading toward as you trace it from either side toward $x = a$. Picture sliding your finger along the curve from the left, then doing the same from the right. If both fingers are heading for the same height on the $y$-axis, that height is the limit.

Keep reading

You've read the first half of Chapter 1. The complete book covers 5 chapters in roughly fifteen pages — readable in one sitting.

Coming soon to Amazon