# Limits and Continuity

## 1.1 Functions and Their Graphs

A good deal of our work with exploring the concept of a limit will be to look at the graphs of functions.  This section will review some of the ideas around how functions and their graphs look, and we will look at two new concepts in functions: step functions and piecewise functions.

Things you should know after today:

• definition of odd and even functions
• what a "one to one" function is
• what the greatest integer function is, and why is it a step function
• how to graph a piecewise function

Resources:

• Notes

Assignment

• p17 #19-24, 32, 34, 43, 45, 47
• p25 #37
• p39 #1-6, 49

## 1.2 Definition of a Limit

Think of a limit as what you think is going to happen, based on what you observe.  Cliffhangers for TV series often work this way; they set up a scene, and make it appear as though something is about to happen, but we don't necessarily know if it is going to go down that way.  A limit is kind of the same thing.  You will observe, graphs, look at the trend or pattern, and then guess what is going to happen at a specific point on the graph without actually know what happens there.  This is a limit.  Find out more about limits and some of the terminology we use with them.

Things you should know after today:

• definition of a limit
• the difference between one-sided and two sided limits
• properties of limits (combining limits
• how to find limits by substituion
• finding limits that can't be evaluated by direct substitution
• finding limits graphically

Resources:

• Notes

Assignment:

• p62 #1-25, 31-43 odds

## 1.3 Limits Involving Infinity

Graphs may have vertical asymptotes, and while we can't see what value the function is going to reach, we can see that the value of the function is becoming infinitely large.  Asymptotes may involve infinity either horizontally or vertically.  We will tie in the concept of asymptotes with limits and end behaviour models today.

Things you should know after today:

• What happens when you get infinite limits at finite values of x
• What happens when you have limits at infinity
• How can you create limits from end behaviour models

Resources

• Notes

Assignment

• p71 #1-21odds 47, 51
• p71 #23-49, 53 (odds)

## 1.5 Limits Involving Trigonometric Functions

A central concept in limits is the sandwich theorem. The sandwich theorem is central in helping us show that

.  You will be using this idea to help you determine limits involving trig functions.

Things you should know after today:

• how to use the sandwich function to determine a limit
• determine limits involving trignometric functions using the properties of limits

Resources

• Notes

Assignment

• p62 #27-30 all, #45-55 odds, #59

## 1.6 Definition of a Continuity

You have explored the concepts of a discrete versus a continuous function in Math 10, but we will be extending those ideas today by exploring what happens when functions are not continuous.  There are several different ways that a function can be discontinuous, and we will be giving names to each kind of discontinuity

Things you should know after today:

• keywords:removable discontinuity, jump discontinuity, infinite discontinuity, oscillating discontinuity
• what the intermediate value theorem says about the range for a particular interval of a function

Resources

• Notes

Assignment

• p80 #1-9 odd, 11-18 all, 19-29 odd, 35-38 all, 41, 42, 45

## 1.7 Average Rate of Change

For any function, you can calculate rates of change on the graph.  There are two different rates of change:

1. average rate of change over an interval
2. instantaneous rate of change at a specifc point or instant in time

The average rate of change can be determined by finding the slope of a secant line using the beginning and ending points of the interval, but to calculate the instantaneous rate of change requires the use of limits.

Things you should know after today:

• difference between a tangent line and a secant line
• how to calculate the slope of a secant line
• how to use the to determine instantaneous rate of change

Resources:

• 1.7 Notes
• 1.8 Solutions (Slopes of Secants and Tangents)

Assignment:

• p87 #1-21 odds

## 1.9 Tangents and Nromals (or Normals)

Tangents are important for finding the instantaneous rate of change for a function at a point or a specific instant.  However, we can also find the equation of a normal line. You will be using your knowledge of secants and tangents to create the equations of tangents and normals to a curve.

Things you should know after today

• what a tangent line is
• what a normal line is
• how to find the equation of a tangent to a curve at a point
• how to find the equation of a normal to curve at a point

Resources:

• Warmup 1.9
• Notes

Assignment

• p88 #23-27, 29, 31, 33, 35