# Limits and Continuity

For in GOD we live, and move, and have our being. - Acts 17:28

The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka

## Welcome to Our Site

I greet you this day,
First: read the notes. Second: view the videos. Third: solve the questions/solved examples. Fourth: check your solutions with my thoroughly-explained solutions. Fifth: check your answers with the calculators as applicable.
The Wolfram Alpha widgets (many thanks to the developers) were used for some calculators.
Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me.
If you are my student, please do not contact me here. Contact me via the school's system.
Thank you for visiting!!!

Samuel Dominic Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S

## Objectives

#### Objectives

Students will:

(1.) Discuss the concept of the limit of a function.

(2.) Discuss the concept of the continuity of a function over a domain.

(3.) Discuss the concept of the discontinuity of a function over a domain.

(4.) Determine the limit of a function numerically.

(5.) Determine the limit of a function algebraically.

(6.) Determine the limit of a function graphically.

(7.) Determine the values of the independent variable for which the dependent variable is continuous.

(8.) Determine the values of the independent variable for which the dependent variable is discontinuous.

(9.) Solve applied problems involving the limit of a function.

(10.) Solve applied problems involving the continuity of a function.

#### Skills Measured/Acquired

(1.) Use of prior knowledge

(2.) Critical Thinking

(3.) Interdisciplinary connections/applications

(4.) Technology

(5.) Active participation through direct questioning

(6.) Research

## Vocabulary Words

Bring it to English: limit, limited, limiting, restriction, continuous, discontinuous,

Bring it to Math:

## Special Limits

I call it Special Limits because of its' importance in the derivatives of certain functions in Calculus.

They include:

$(1.)\:\: \displaystyle{\lim_{\theta \to 0}} \dfrac{\sin\theta}{\theta} \\[5ex] \theta \:\:is\:\:in\:\:Radians \\[3ex]$ We shall determine this limit both numerically and algebraically

Determining $\boldsymbol{\displaystyle{\lim_{\theta \to 0}} \dfrac{\sin\theta}{\theta}}$ Numerically

 $\theta$(RAD) $-0.1$ $-0.01$ $-0.001$ $-0.0001$ $-0.00001$ $0$ $0.00001$ $0.0001$ $0.001$ $0.01$ $0.1$ $\sin\theta$ $-0.09983$ $-0.00999983$ $-0.00099999983$ $-0.00009999999983$ $-0.00001$ $0.00001$ $0.00009999999983$ $0.00099999983$ $0.00999983$ $0.09983$ $\dfrac{\sin\theta}{\theta}$ $0.09983$ $0.999983$ $0.99999983$ $0.9999999983$ $1$ $\boldsymbol{1}$ $1$ $0.9999999983$ $0.99999983$ $0.999983$ $0.09983$

Determining $\boldsymbol{\displaystyle{\lim_{\theta \to 0}} \dfrac{\sin\theta}{\theta}}$ Algebraically

### References

Chukwuemeka, S.D (2016, April 30). Samuel Chukwuemeka Tutorials - Math, Science, and Technology. Retrieved from https://www.samuelchukwuemeka.com

Stroud, K. A., & Booth, D. J. (2001). Engineering Mathematics ($5^{th}$ ed.). Basingstoke: Palgrave.

Tan, S. T. (2004). Applied Calculus for the Managerial, Life, and Social Sciences ($5^{th}$ ed.). Pacific Grove, CA: Brooks/Cole Publishing Company.

Waner, S., & Costenoble, S. (2013). Applied Calculus ($6^{th}$ ed.). Boston, MA: Cengage Learning.