1

§ 1-4 Limits, Continuity, and Intermediate Value Thereom

The student will learn about:limits,

infinite limits,

and continuity.

limits, finding limits, one-sided limits,

Limits

The word “limit” is used in everyday conversation to describe the ultimate behavior of something, as in the “limit of one’s endurance” or the “limit of one’s patience.”

In mathematics, the word “limit” has a similar but more precise meaning.

Limits

Given a function f(x), if x approaching 3 causes the function to take values approaching (or equalling) some particular number, such as 10, then we will call 10 the limit of the function and write

In practice, the two simplest ways we can approach 3 are from the left or from the right.

Limits

For example, the numbers 2.9, 2.99, 2.999, ... approach 3 from the left, which we denote by x→3 –, and the numbers 3.1, 3.01, 3.001, ... approach 3 from the right, denoted by x→3 +. Such limits are called one-sided limits.

Use tables to find

Example 1 – FINDING A LIMIT BY TABLES

Solution :

We make two tables, as shown below, one with x approaching 3 from the left, and the other with x approaching 3 from the right.

6

Limits IMPORTANT!This table shows what f (x) is doing as x approaches 3. Or we have the limit of the function as x approaches We write this procedure with the following notation.

x 2 2.9 2.99 2.999 3 3.001 3.01 3.1 4

f (x) 8 9.8 9.98 9.998 ? 10.002 10.02 10.2 12

x 3lim 2x 4 10

Def: We write

if the functional value of f (x) is close to the single real number L whenever x is close to, but not equal to, c. (on either side of c).

or as x → c, then f (x) → L

3

10

H

x clim f (x) L

Limits

As you have just seen the good news is that many limits can be evaluated by direct substitution.

8

Limit PropertiesThese rules, which may be proved from the definition of limit, can be summarized as follows.

For functions composed of addition, subtraction, multiplication, division, powers, root, limits may be evaluated by direct substitution, provided that the resulting expression is defined.

cx

f (x) )cf (lim

Examples – FINDING LIMITS BY DIRECT SUBSTITUTION

x 4

1. xlim

Substitute 4 for x.4 2

2

x 6

x2.

x 3lim

26 364

6 3 9

Substitute 6 for x.

10

Example

But be careful when a quotient is involved.

2

x 2

x x 6 0lim Which is undefined!

x 2 0

2

x 2 x 2 x 2

x x 6 (x 3)(x 2)lim lim lim (x 3) 5

x 2 x 2

2x x 6

NOTE : f ( x ) graphs as a straight line.x 2

Graph it.

But the limit exist!!!!

What happens at x = 2?

11

One-Sided Limit

We have introduced the idea of one-sided limits. We write

and call K the limit from the left (or left-hand limit) if f (x) is close to K whenever x is close to c, but to the left of c on the real number line.

K)x(flimcx

5

12

One-Sided Limit

We write

and call L the limit from the right (or right-hand limit) if f (x) is close to L whenever x is close to c, but to the right of c on the real number line.

L)x(flimcx

13

The Limit

K)x(flimcx

Thus we have a left-sided limit:

L)x(flimcx

And a right-sided limit:

And in order for a limit to exist, the limit from the left and the limit from the right must exist and be equal.

14

Examplef (x) = |x|/x at x = 0

x 0

xlim 1

x

The left and right limits are different, therefore there is no limit.

0x 0

xlim 1

x

15

Infinite Limits

Sometimes as x approaches c, f (x) approaches infinity or negative infinity.

2x 2

1lim

x 2 Consider

From the graph to the right you can see that the limit is ∞. To say that a limit exist means that the limit is a real number, and since ∞ and - ∞ are not real numbers means that the limit does not exist.

16

Intro to ContinuityAs we have seen some graphs have holes in them, some have breaks and some have other irregularities. We wish to study each of these oddities.

We will use our information of limits to decide if a function is continuous or has holes.

Continuity

Intuitively, a function is said to be continuous if we can draw a graph of the function with one continuous line. I. e. without removing our pencil from the graph paper.

18

Definition

A function f is continuous at a point x = c if

x clim f (x) exists

2.

f (c) is defined 1.

x clim f (x) f (c)

3.

THIS IS THE DEFINITION OF CONTINUITY

19

Examplef (x) = x – 1 at x = 2.

x 2f (2) 1 lim x 1

c.

x 2lim x 1 1

b.The limit exist!

f (2) =a. 1

Therefore the function is continuous at x = 2.

2

1

20

Example

f (x) = (x2 – 9)/(x + 3) at x = -3

2

x 3

x 9lim f ( 3)

x 3

c.

2

x 3

x 9lim

x 3

b. - 6

The limit exist!

f (-3) = 0/0a. Is undefined!

Therefore the function is not continuous at x = -3.

-3

-6

You can use table on your calculator to verify this.

21

Continuity Properties

If two functions are continuous on the same interval, then their sum, difference, product, and quotient are continuous on the same interval except for values of x that make the denominator 0.

Every polynomial function is continuous. Every rational function is continuous except where the denominator is zero.

22

Continuity Summary.

3. We have discontinuity with some functions that have a gap.

Graph on your calculator with a standard window.

Functions have three types of discontinuity. Consider - 2

2

x 4x 5f ( x )

x 2x 15

1. Discontinuity at vertical asymptote.

2. Discontinuity at hole.

23

Summary.

• We learned about left and right limits.

• We learned about limits and their properties.

• We learned about continuity and the properties of continuity.

Intermediate Value Theorem

Section 1.4

25

Intermediate Value Theorem: Intuition

• Traveling on France’s TGV trains, you reach speed of 280 mi/hr.

• How do you know at some point of train ride you were traveling 100 mi/hr?

• To go from 0 to 280, must have passed through 100 mi/hr since speed of train changed continuously

26

Intermediate Value Theorem

• Suppose that f is continuous on the closed interval [a,b]. If L is any real number between f(a) and f(b) then there must be at least one number c on the open interval (a,b) such that f(c) = L.

27

28

Limitations of IVT• If d(0) = 100 and d(10) = 35, where t is measured in seconds. • d is a continuous function, the IVT tells you that at some point

between t=0 and t =10, the decibel level reached every value between 35 and 100.

• It does NOT say anything about: • When or how many times (other than at least once) a particular

decibel was attained. • Whether or not decibel levels bigger than 100 or less than 35

were reached.

29

The Difference Between VROOOOOOOOM and VROOOOOOOM.

These graphs of PC's noise illustrate that very different behaviors are consistent with the hypothesis that d(t) is continuous and that its values at t=0 and t=10 are 100 and 35 respectively.

30

Example 1:

• Sketch a graph to decide if the cosecant function, f(x) = csc (x) is continuous over the domain [-π, π].

31

Example 2

• Consider the equation sin x = x – 2 . Use the intermediate Value Theorem to explain why there must be a solution between π/2 and π.

Example 3

• Consider the function ,

• Calculate f(6), f(-5.5), f(0)

• Can you conclude that there must be a zero between f(6) and f(-5.5)?