Make a Stick Activity

• Use markers/ colored pencils/ or whatever you like to write your name on a stick and decorate it to make it your own.

• When you’re done, bring your stick up to the front table and put it in the box for your class.

Welcome To Calculus BCExpectations• Effort• Ask Questions – All are valid• RESPECT- Support and help each other• Return books to back shelf• Bring: notebook, pencil, red pen, calculator• Backpacks & purses under desk or in back of room• Cell phones on silent and not on your person

My teacherpage: http://www.husd.org//Domain/1454

AP Expectations• This class is an AP course. Everyone

takes the AP test in May.

• You will need to spend a significant amount of time outside of class on homework, AP review, and preparation for the AP test.

Limits and Their Properties1

Copyright © Cengage Learning. All rights reserved.

BC Day 1

Limits Review

Handout: Limits and their Properties Notes

Copyright © Cengage Learning. All rights reserved.

1.2-1.4A

Estimate limits numerically, graphically and algebraically.

Learn different ways that a limit can fail to exist.

Special Trig Limits

Define Continuity.

Objectives

Formal definition of a Limit:

If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit of f(x), as x approaches c, is L.

“The limit of f of x as x approaches c is L.”

limx c

f x L

This limit is written as

Consider the graph of f(θ) = sin(θ)/θ

• Let’s fill in the following table

• We can say that the limit of f(θ) approaches 1 as θ approaches 0 from the right

• We write this as

• We can construct a similar table to show what happens as θ approaches 0 from the left

θ 0.5 0.4 0.3 0.2 0.1 0.05

sin(θ)/θ

1sin

lim0

θ 0.5 0.4 0.3 0.2 0.1 0.05

sin(θ)/θ 0.959 0.974 0.985 0.993 0.998 0.9995

θ -0.5 -0.4 -0.3 -0.2 -0.1 -0.05

sin(θ)/θ 0.959 0.974 0.985 0.993 0.998 0.9995

• So we get

• Now since we have we say that the limit exists and we write

1sin

lim0

1sin

limsin

lim00

1sin

lim0

3

1

1lim

1x

x

x

An Introduction to Limits

Ex: Find the following limit:

Start by sketching a graph of the function

For all values other than x = 1, you can use standard

curve-sketching techniques.

However, at x = 1, it is not clear what to expect.

We can find this limit numerically:

An Introduction to Limits

To get an idea of the behavior of the graph of f near x = 1, you can use two sets of x-values–one set that approaches 1 from the left and one set that approaches 1 from the right, as shown in the table.

An Introduction to Limits

The graph of f is a parabola that has a gap at the point (1, 3), as shown in the Figure 1.5.

Although x can not equal 1, you can move arbitrarily close to 1, and as a result f(x) moves arbitrarily close to 3.

Using limit notation, you can write

An Introduction to Limits

This is read as “the limit of f(x) as x approaches 1 is 3.”

Figure 1.5

This discussion leads to an informal definition of a limit:

A limit is the value (meaning y value) a function approaches as x approaches a particular value from the left and from the right.

Properties of Limits:

Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power.

For a limit to exist, the function must approach the same value from both sides.

One-sided limits approach from either the left or right side only.

x clim f x L

x clim f x L

x clim f x L

1 2 3 4

1

2

y f x

limx

f x

Limits That Fail to Exist - 3 Reasons

Discuss the existence of the limit:

Solution: Using a graphical representation, you can see that x does not approach any number. Therefore, the limit does not exist. 0

limx

20

1limx x

0limx

0limx

DNE

Properties of Limits

Properties of Limits

Properties of Limits

Compute the following limits

2

1lim

1x x

3

3lim

1x

x

x

sin 2limx

x

x

21

1lim

2x

x

x x

4

lim tanx

x

lim cosx

x x

• Let’s take a look at the last one

• What happened when we plugged in 1 for x?

• When we get we have what’s called an

indeterminate form

• Let’s see how we can solve it

2

1lim

21

xx

xx

0

0

• Let’s look at the graph of

Is the function continuous at x = 1?

2

1lim

21

xx

xx

Strategies for Finding Limits?

You Try:

Find the limit:

5

You Try:

3

3

27Evaluate lim

3x

x

x

27

Find the limit:

Solution:

By direct substitution, you obtain the indeterminate form 0/0.

Example – Rationalizing Technique

In this case, you can rewrite the fraction by rationalizing the numerator.

cont’dSolution

Now, using Theorem 1.7, you can evaluate the limit as shown.

cont’dSolution

A table or a graph can reinforce your conclusion that the

limit is . (See Figure 1.20.)

Figure 1.20

Solutioncont’d

Solutioncont’d

You Try:

0

4 2Evaluate lim

x

x

x

1

4

Example:

0

2

Evaluate lim

1, 0

1, 0

x

f x

x xf x

x x

0

limx

f x

0

limx

f x

1 1

SPECIAL TRIG LIMITS

You must know these for the AP test!

Find the limit:

Solution:Direct substitution yields the indeterminate form 0/0.

To solve this problem, you can write tan x as (sin x)/(cos x) and obtain

Example – A Limit Involving a Trigonometric Function

Solution cont’d

Now, because

you can obtain

Figure 1.23

Solution cont’d

AP example

• Find the following:

0

sin 4x

xLim

x

0

sinlim 1x

x

x

4

Continuity at a Point and on an Open Interval

Figure 1.25 identifies three values of x at which the graph of f is not continuous. At all other points in the interval (a, b), the graph of f is uninterrupted and continuous.

Figure 1.25

Continuity at a Point and on an Open Interval

Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil.

A function is continuous at a point if the limit is the same as the value of the function.

This function has discontinuities at x=1 and x=2.

It is continuous at x=0 and x=4, because the one-sided limits match the value of the function1 2 3 4

1

2

Consider an open interval I that contains a real number c.

If a function f is defined on I (except possibly at c), and f is not continuous at c, then f is said to have a discontinuity at c. Discontinuities fall into two categories: removable and nonremovable.

A discontinuity at c is called removable if f can be made continuous by appropriately defining (or redefining f(c)).

Continuity at a Point and on an Open Interval

For instance, the functions shown in Figures 1.26(a) and (c) have removable discontinuities at c and the function shown in Figure 1.26(b) has a nonremovable discontinuity at c.

Figure 1.26

Continuity at a Point and on an Open Interval

Example 1 – Continuity of a Function

Discuss the continuity of each function.

Example 1(a) – Solution

Figure 1.27(a)

The domain of f is all nonzero real numbers.

From Theorem 1.3, you can conclude that f is continuous at every x-value in its domain.

At x = 0, f has a non removable discontinuity, as shown in Figure 1.27(a).

In other words, there is no way to define f(0) so as to make the functioncontinuous at x = 0.

The domain of g is all real numbers except x = 1.

From Theorem 1.3, you can conclude that g is continuous at every x-value in its domain.

At x = 1, the function has a removable discontinuity, as shown in Figure 1.27(b).

If g(1) is defined as 2, the “newly defined” function is continuous for all real numbers.

Figure 1.27(b)

cont’dExample 1(b) – Solution

Figure 1.27(c)

Example 1(c) – Solution

The domain of h is all real numbers. The function h is continuous on and , and, because , h is continuous on the entire real line, as shown in Figure 1.27(c).

cont’d

The domain of y is all real numbers. From Theorem 1.6, you can conclude that the function is continuous on its entire domain, , as shown in Figure 1.27(d).

Figure 1.27(d)

Example 1(d) – Solutioncont’d

Removing a discontinuity:

3

2

1

1

xf x

x

has a discontinuity at 1x

Write an extended function that is continuous at 1x

3

21

1lim

1x

x

x

2

1

1 1lim 1 1x

x x xx x

1 1 1

2

3

2

3

2

1, 1

13

, 12

xx

xf x

x

Note: There is another discontinuity at that can not be removed.1x

Removing a discontinuity:

3

2

1, 1

13

, 12

xx

xf x

x

Group Work:

Sketch the graph of f. Identify the values of c for which exists. lim

x cf x

lim exists for all values except where 4.x c

f x c

Homework

Limits and Continuous Functions WS

Get Books! This ppt is on my teacher-page.