# 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 calculus, the squeeze theorem is a principle that examines the limit of a function. Explore this concept in theory and in practice through examples.

### 2. Using a Graph to Define Limits

Limits can be visualized and defined better with the use of graphs. Explore examples of limits defined by graphs such as speed limits and limits of a pendulum, as well as other limits in math.

### 3. Understanding Limits: Using Notation

Limits describe what happens as you get closer to a number. Learn about defining and understanding limits. Then, learn about finding limits, using notation.

### 4. One-Sided Limits and Continuity

When studying mathematics functions and methodology of calculation, a good place to start is understanding the significance of one-sided limits and continuity. Learn more about the properties and functions, and study an example of a formula for finding one-sided limits and continuity.

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

A limit can tell us the value that a function approaches as that function's inputs get closer and closer to a number. Learn more about how to determine the limits of functions, properties of limits and read examples.

### 6. Understanding the Properties of Limits

The key to understanding the properties of limits is that it requires dividing the function into its individual parts first. Learn more about the properties of limits, including the addition and subtraction property, product property, and division property.

### 7. Continuity in a Function

Continuity is the state of an equation or graph where the solutions form a continuous line, with no gaps on the graph. Learn the concept of continuity, opposed by discontinuity, and examples of both types of functions.

### 8. Discontinuities in Functions and Graphs

A discontinuity is where the potential values in an equation 'jump', rather than being continuous as with an un-broken line on a graph. See how discontinuities appear in graphs and equations, including jump discontinuities and asymptotic discontinuities.

### 9. Regions of Continuity in a Function

A region of continuity is where you have a function that is continuous and is a critical understanding in calculus and mathematics. Learn more about regions of continuity as a function and read examples.

### 10. Continuous Functions Theorems

Continuous functions are functions that have their conditions satisfied between multiple points, appearing as an uninterrupted line when graphed. See examples of how this is represented in the intermediate value theorem and the extreme value theorem.

### 11. Intermediate Value Theorem: Definition

The intermediate value theorem concerns the properties of continuous functions over a range and finding at least one solution. Learn more about the definition of the intermediate value theorem using the altitudes of a jet airplane as an example.

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

The intermediate value theorem states that a function, when continuous, can have a solution for all points along the range that it is within. Identify the applications of this theorem in finding roots, and solve some example functions.

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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