Limits
Introduction to limits and continuity
Start with the basics and progress through 10 lessons. Each lesson builds on the previous one.
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10 questions, new mix each time (from 178)
In This Topic (10 lessons)
Introduction to Limits
Learn the foundational concept of limits and how they describe the behavior of functions as inputs approach specific values.
Limit Notation
Master the symbolic language of limits and learn how to read, write, and interpret limit expressions.
Evaluating Limits Graphically
Read limits straight from a graph, tell the limit apart from the function value, and spot when a limit does not exist.
Evaluating Limits from Tables
Estimate a limit numerically: build a table of values from both sides, move closer and closer, and read where the outputs are heading.
Limits by Direct Substitution
Find limits algebraically: plug in the value when you can, and when you get 0/0, factor and cancel or rationalize first.
One-Sided Limits
Learn to find left-hand and right-hand limits from graphs and piecewise formulas, and decide when a two-sided limit exists.
Limits at Infinity
Find what a function approaches as x grows without bound, compare degrees of rational functions and connect limits at infinity to horizontal asymptotes.
Infinite Limits
Learn what it means when a function grows without bound near a point, how to find the sign with denominator analysis, and how infinite limits create vertical asymptotes.
Continuity
Learn the three-part definition of continuity, where common functions are continuous, how to find a constant that makes a piecewise function continuous, and the Intermediate Value Theorem.
Types of Discontinuity
Learn to tell removable discontinuities (holes), jump discontinuities and infinite discontinuities apart, from graphs and from formulas.
Limits describe the behavior of functions as inputs approach certain values. This foundational concept is the gateway to calculus, enabling the study of derivatives and integrals.
What You'll Learn
- Evaluate limits graphically
- Evaluate limits algebraically
- Understand one-sided limits
- Identify when limits do not exist
- Apply limit laws
Frequently Asked Questions
What is a limit?
The value a function approaches as x approaches a certain value. It describes behavior near a point, not at the point.
When does a limit not exist?
When left and right limits are different, or the function oscillates or goes to infinity.
What are the limit laws?
Limits of sums, differences, products, and quotients can be found from limits of individual functions (when they exist).