Limits and Continuity

There are 3 main questions that will drive this unit. They are:

1. Can change occur at an instant?

2. How does knowing the value of a limit, or that a limit does not exist, help you to make sense of interesting features of functions and their graphs?

3. How do we close loopholes so that a conclusion about a function is always true?

Limits introduce the subtle distinction between evaluating a function at a point and considering what value the function is approaching, if any, as x approaches a point. This distinction allows us to extend understanding of asymptotes and holes in graphs with formal definitions of continuity.

By the end of the unit, students will be able to do the following:

• Discuss the idea of a limit
• Investigate a limit using a table and/or a graph
• Find the limit of a

• sum
• difference
• product
• power
• root
• polynomial
• quotient
• average rate of change
• difference quotient
• Determine whether a function is continuous at a number
• Determine intervals on which a function is continuous
• Use properties of continuity
• Use the Intermediate Value Theorem (IVT)
• Use the Squeeze Theorem to find a limit
• Find limits involving trigonometric functions
• Determine where the trigonometric functions are continuous
• Determine where an exponential or a logarithmic functions is continuous
• Investigate infinite limits
• Find the vertical asymptotes of a graph
• Investigate limits at infinity
• Find the horizontal asymptotes of a graph
• Find the asymptotes of the graph of a rational function

The concept of a limit is central to calculus. To understand calculus, it is essential to know what it means for a function to have a limit, and then how to find a limit of a function.

What does it mean for a function to have a limit as x approaches some fixed number c?

This is the question that is what drives limits.

You can use tables to have a better understanding of limits.

When we look at a limit, in order for a limit to exist, we must approach the same number from the right and from the left. If you are approaching the same number from both the left and the right, then the limit, as a whole will exist. When you are specifically looking at approaching a limit from the left side and/or the right side, these limits are called one-sided limits because you are only looking at approaching a limit from one particular side. These are denoted by the following:

When we deal with limits, we do not worry about if the point is actually defined at x, we just want to know what number are we actually approaching. In a table, you may or may not be able to see that, but in the graph you can. The following 3 pictures show all valid limits.

The limit L of a function y = f(x) as x approaches a number x does not depend on the value of f at c.

The limit L of a function y = f(x) as x approaches a number c is unique, that is, a function cannot have more than one limit.

If there is no single number that the value of f approaches as x gets close to c, we say that f has no limit as x approaches c, or more simply, that the limit of f does not exist at c.

In order for a limit to fully exist, the following must be true:

A function is continuous at a number c if the following three conditions are met:

1. f(c) is defined (that is, c is in the domain of f(x))

2. The limit as x approaches c of f(x) exists

3. The limit as x approaches c of f(x) is equal to f(c)

When looking to see if a number is continuous, you must look at the both the left and the right side of the number, c. If the interval is (a, b) and c is between the interval, then we look at the following.

1. A function is continuous on an open interval (a, b) if f(x) is continuous at every number in (a, b)

2. A function is continuous on an interval [a, b) if f(x) is continuous on the open interval (a, b) and continuous from the right at the number a.

3. A function is continuous on an interval (a, b] if f(x) is continuous on the open interval (a, b) and continuous from the left at the number b.

4. A function is continuous on an interval [a, b] if f(x) is continuous on the open interval (a, b) and continuous from the left at the number b and from the right at the number a.

A function is continuous on its domain if it is continuous at every number c in its domain.

• A polynomial function is continuous on its domain, all real numbers.

• A rational function is continuous on its domain if
• The graph has no holes or gaps on that interval
• The function is continuous on its domain, it will be continuous at every number in its domain. The graph itself may have holes or gams at numbers that are not in the domain.

• If the functions f(x) and g(x) are continuous at a number c, and if k is a real number, then the functions

f(x) + g(x)

f(x) - g(x)

f(x) * g(x)

kf(x)

are also continuous at c.

• If g(c) does not equal zero, then the function f(x)/g(x) is continuous at c.

• If a function g(x) is continuous at c and a function f(x) is continuous at g(c), then the composite function f(g(x)) is continuous at c. That is:

• If f(x) is a one to one function that is continuous on it domain, then its inverse function is also continuous on its domain.

This is the first of 4 big theorems that will be tested on the AP exam.

Suppose f(x) is a function that is continuous on a closed interval [a, b] and that f(a) does not equal f(b). If N is any number between f(a) and f(b), then there is at least one number c in the open interval (a, b) for which f(c) = N.

In other words, a function, f(x), that is continuous on a closed interval [a, b] must take on all values between f(a) and f(b).

The Intermediate Value Theorem is an existence theorem. It states that there is at least one number c for which f(C) = N, but it does not tell us how to find c. That part will be left up to us.

In Pre-Calculus, you were first introduced to the Intermediate Value Theorem to determine if there was at least one zero on the interval (a, b). You did this by looking to see if f(a) and f(b) had a sign change. If f(a) was positive and and f(b) was negative, then there was at least one zero on that interval (a, b). The reverse is also true.

To use the Squeeze Theorem to find the limit of g(x), we need to know, or be able to find, two functions f(x) and h(x) that "sandwich" the function g(x) between them for all of x close to c.

Here are the limits that you need to know.

The line x = a is called a vertical asymptote of f(x) if at least one of the following it true

The line y = L is a horizontal asymptote of f(x) is either of the following is true:

Remember from Pre-Calculus and Algebra 2, when you look for horizontal asymptotes, you have to look at the end behavior of the graph. There are 3 options for horizontal asymptotes:

In terms of limits, the three options are the following:

Unit 1

Due September 6th, 2024

Section 1.1

1, 2, 9, 13, 15, 19, 21, 25, 31, 35, 37, 41, 55, 57, All AP Practice Problems

Section 1.2

13, 15, 19, 23, 29, 31, 33, 35, 41, 47, 51, 55, 59, 63, 69, 71, 73, 79, 81, All AP Practice Problems

Section 1.3

13, 15, 19, 21, 25, 29, 35, 37, 45, 47, 53, 59, 63, 65, 77, All AP Practice Problems

Section 1.4

5, 9, 15, 19, 23, 25, 31, 33, 35, 39, 45, 47, All AP Practice Problems

Section 1.5

9-16, 27, 31, 37, 41, 43, 45, 47, 49, 57, 59, 63, 69, 79, All AP Practice Problems

Page 159

All AP Practice Problems