Limits

Section 15-1

WHAT YOU WILL LEARN:

1. How to calculate limits of polynomial and rational functions algebraically.

2. How to evaluate limits of functions using a calculator.

Yippeeeee!! Calculus

• History- Early signs in ancient Egypt, India and Greece.

- Sir Isaac Newton and Gottfried Leibniz are usually credited with the invention of modern calculus in the late 17th century. Their most important contributions were the development of the fundamental theorem of calculus. Also, Leibniz did a great deal of work with developing consistent and useful notation and concepts. Newton was the first to organize the field into one consistent subject, and also provided some of the first and most important applications, especially of integral calculus. * Wikipedia

More History

• Two branches of Calculus:

- Differential = deals with variable or changing quantities.

- Integral = deals with finding sums of infinitesimally small quantities.

Today we will talk about “Limits”

An Example from Sports

Penalty 1st 2nd 3rd

Yard Line 5 2.5 1.25

• In football, if the length of a penalty exceeds half the distance to the offending team’s goal line then the ball is moved only half the distance to the goal line. Suppose one team has the ball at the other team’s 10 yard line. The other team, in an effort to prevent a touchdown, repeatedly commits penalties. After the first penalty, the ball would be moved to the 5 yard line. If this continues you would see:

Vocabulary

)(lim xfLax

• A number that the terms of a sequence approach, without necessarily reaching it, is called a limit.

• Official definition: If there is a number L such that the value of f(x) gets closer and closer to L as x gets closer to a number a, then L is called the limit of f(x) as x approaches a.

• In symbols,

Method 1: Evaluating Limits by Looking at Graphs

)(lim2

xfx

• First, we will look at evaluating limits by looking at graphs.

• Consider the graph of the function y = f(x) shown below. Find the following:

A. f(2) and

B. f(4) and

x-10 -5 5 10

y

-10

-5

5

10

)(lim4

xfx

You Try

x-10 -5 5 10

y

-10

-5

5

10

• Consider the graph of the function y = f(x). Find the following:

• A. f(-2) and

• B. f(2) and

)(lim2xf

x

)(lim2

xfx

Remember Continuity

)()(lim afxfax

• Sometimes the value of a function at a given value of x and the limit of the function as it approaches that value may be different (as in the previous two examples).

• Continuity: In order for a function to be continuous:

• In other words…no gaps

Method 2: Evaluating a Limit Algebraically

)1075(lim 23

3

xxx

x

• Evaluating a limit algebraically when the function is continuous at the given value of x. Example:

• Evaluate each limit.

A.

B. x

xx

coslim

You Try

)101062(lim 23

2

xxx

x

• Evaluate each limit.

A.

B. x

xx

3sinlim

Method 3: Evaluating a Limit When Function is not Continuous

3

9)(lim

2

3

x

xxf

x

• When a function is not continuous at a given value, it is more difficult to evaluate the limit at that value. Example:

Evaluate the limit:

Let’s look at that on a graphing calculator.

More Examples of Method 3

xx

xxx 4

82lim

2

2

4

• Evaluate each limit.

A.

B. h

hhhh

64lim

23

0

You Try

4

2lim

2

2

2

x

xxx

Evaluate each limit.

1.

2. 1

1lim

2

4

1

x

xx

Method 4: Using a Table• Sometimes you can’t simplify an expression in

such a way that you can get the limit algebraically. At that point, you can resort to an x/y table of values that use values of x that get increasingly close (but never quite to) the given value of x.

• Like proving continuity in Chapter 3.

An Example

x

xx

sinlim

0

• What would happen if you tried to evaluate the limit as x approaches zero for the following function:

• Can you evaluate directly (by plugging in)?

• Can you simplify?

Evaluating the Limit

x

xx

sinlim

0

Step 1: Adjust your calculator!

Step 2: Put in values for x that get closer and closer to zero, approaching from both sides.

Step 3: See if they you are approaching the same value from both sides. What is it?

More Examples

20

cos1lim

x

xx

• Evaluate each limit.

A.

B. 1

lnlim

1 x

xx

You Try

x

xx

1coslim

0

Evaluate each limit.

1.

2. x

xx sin

)1ln(lim

0

So…When Do I Use Which Method• If you only have a graph, use method 1 (looking at

the graph).

• If you have a function that can be evaluated directly, use method 2 (plugging limit back into function).

• If the function can be simplified, use method 3 (simplify then plug the limit back in).

• If the limit can’t be evaluated directly and can’t be simplified, use method 4 (use your calculator and tables).

Homework

page 946, 12-14 all, 16, 18, 20-28 even, 36, 38