# Ch 5: Limits of Sequences & Functions

### About This Chapter

## Limits of Sequences & Functions - Chapter Summary

Functions are a vital part of precalculus mathematics. Our experienced instructors teach you the limits of sequences, including convergent and divergent sequences. You'll explore the left- and right-hand limits of functions, as well as one-sided limits. Work with continuity and discontinuity in functions and graphs, and learn to apply theorems when solving precalculus math problems.

When you have completed each of these lessons explaining limits of sequences and functions, you should understand:

- Squeeze theorem
- How to use notation and define limits using a graph
- Limits of functions and their properties
- Continuity and discontinuities in functions and graphs
- Regions of continuity
- Continuous function theorems
- Intermediate value theorem

These engaging but brief video lessons thoroughly explain limits of sequences and functions. If you have any questions as you work, just ask our subject experts. Video tags throughout the lessons make it easy to review examples or formulas as you study, and the quizzes at the end of each lesson will show you just how much you have learned.

### 1. Squeeze Theorem: Definition and Examples

In the Kingdom of Rimonn there are three rivers. In this lesson, learn how these waterways demonstrate the power of the squeeze theorem for finding the limits of functions.

### 2. Using a Graph to Define Limits

My mom always said I tested the limits of her patience. Use graphs to learn about limits in math. You won't get grounded as we approach limits in this lesson.

### 3. Understanding Limits: Using Notation

Join me on a road trip as we define the mathematical notation of limits. As time goes by and I traverse hills and highways, the limit of my speed changes. Learn how to write these limits in this lesson.

### 4. One-Sided Limits and Continuity

Over the river and through the woods is only fun on a continuous path. What happens when the path has a discontinuity? In this lesson, learn about the relationship between continuity and limits as we walk up and down this wildlife path.

### 5. How to Determine the Limits of Functions

You know the definition of a limit. You know the properties of limits. You can connect limits and continuity. Now use this knowledge to calculate the limits of complex functions in this lesson.

### 6. Understanding the Properties of Limits

Graphically we can see limits, but how do we actually calculate them? Three words: Divide and Conquer. In this lesson, explore some of the properties that we can use to find limits.

### 7. Continuity in a Function

Travel to space and explore the difference between continuous and discontinuous functions in this lesson. Learn how determining continuity is as easy as tracing a line.

### 8. Discontinuities in Functions and Graphs

In this lesson, we talk about the types of discontinuities that you commonly see in functions. In particular, learn how to identify point, jump and asymptotic discontinuities.

### 9. Regions of Continuity in a Function

Can Earth ever compete with extraterrestrial UFOs? In this lesson, you'll learn that not all functions are continuous, but most have regions where they are continuous. Discover how to define regions of continuity for functions that have discontinuities.

### 10. Continuous Functions Theorems

In this lesson, you'll learn the basics of continuous functions. Then, we'll look at two theorems pertaining to these functions: the intermediate value theorem and the extreme value theorem.

### 11. Intermediate Value Theorem: Definition

A UFO and a jet take off and ascend to 30,000 feet along discontinuous and continuous paths, respectively. In this lesson, learn about the intermediate value theorem and why the jet has to cross 15,000 feet.

### 12. Intermediate Value Theorem: Examples and Applications

Many problems in math don't require an exact solution. Some problems exist simply to find out if any solution exists. In this lesson, we'll learn how to use the intermediate value theorem to answer an age-old question.

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### Other Chapters

Other chapters within the GRE Math: Study Guide & Test Prep course

- Functions in Precalculus
- Analytical Geometry in Precalculus
- Polynomial Equations in Precalculus
- Logarithms & Trigonometry
- Calculating Derivatives
- Curve Sketching in Precalculus
- Differentiable Functions & Min-Max Problems
- Indefinite Integrals in Calculus
- Definite Integrals in Calculus
- Additional Topics in Calculus
- L'Hopital's Rule, Integrals & Series in Calculus
- Analytic Geometry in 3-Dimensions
- Partial Derivatives
- Calculus: Min/Max & Integrals
- Algebra: Differential Equations
- Algebra: Matrices & Vectors
- Algebra: Determinants & Transformations
- Algebra: Number Theory & Abstract Algebra
- Additional Topics: Sets
- Additional Topics: Unions & Intersections
- Additional Topics: Graphing & Probability
- Additional Topics: Standard Deviation
- Additional Topics: Topology & Complex Variables
- Additional Topics: Trigonometry
- Additional Topics: Theorems, Analysis & Optimizing
- GRE Math Flashcards