LIMITS CONT.Evaluating them-

Numerically, Anallyitically and Graphically

1.3 PROPERTIES OF LIMITS:

limx cb b

1lim5 5x

limx cx c

lim n n

x cx c

2lim 2xx

3 3

4lim 4xx

64

PROPERTIES: Scalar multiple

Sum or difference

Product

Quotient

Power

let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:

let b = 2, c = 5, f(x) = x and g(x)=x2

SCALAR MULTIPLE :

limx cf x L

lim

x cg x M

limx cbf x bL

5lim 2x

x

2 5 10

let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:

let b = 2, c = 5, f(x) = x and g(x)=x2

SUM OR DIFFERENCE:

limx cf x L

lim

x cg x M

lim limx c x cf x g x L M

2

5 5lim limx xx x

5 25 30

let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:

let b = 2, c = 5, f(x) = x and g(x)=x2

PRODUCT:

limx cf x L

lim

x cg x M

lim limx c x cf x g x L M

2

5 5lim limx xx x

5 25 125

let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:

let b = 2, c = 5, f(x) = x and g(x)=x2

QUOTIENT:

limx cf x L

lim

x cg x M

limx c

f x Lg x M

2limx c

xx

5 125 5

( ),M 0g x

let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:

let b = 2, c = 5, f(x) = x and g(x)=x2

POWER:

limx cf x L

lim

x cg x M

limn n

x cf x L

3

5limx

x

35 125

Functions can all be combined to form more complex functions:

We can also look at this another way:

Sometimes this makes it easier to calculate… we will deal with this more in the future.

COMPOSITION OF FUNCTIONS:

2 4 f x x g x x

0

lim ?xg f x

2

0lim 4 0 4 2x

x

2

0lim 4 0 4 2x

x

TRIGONOMETRIC LIMITS: Let c be a real number in the domain of

the given trigonometric function:

This works this way for all of the trig functions.

limsin sinx c

x c

TRIGONOMETRIC LIMITS:

0lim tanx

x

tan 0 0

lim cosxx x

cos

2

4

lim sinx

x

2

4

lim sinx

x

2

4

lim sinx

x

2

sin4

22 1

2 2

FINDING LIMITS:DIVIDING OUT TECHNIQUEConsider

Direct substitution yields an indeterminate form.Graph it. What does the function approach as x gets closer to 2?What about a table? What does it look like it is approaching?Can we do algebra? (see next slide for steps)

What answer does that give us?

3

2

8lim2x

xx

USE ALGEBRA: Find: Use :

So, = = 12

SOLVING LIMITS: direct substitution.

Simplify using algebra and then try direct substitution again.

Use a graph or a table to reinforce your conclusion or to evaluate the limit if you are allowed to use technology.

SQUEEZE THEOREM: Not necessary for AP to be added later

DEFINITION Formal Definition:

Epsilon-deltaThis formal definition is rather intimidating when you first look at it, but when broken down it makes sense.

ONE-SIDED LIMITS: Lets look at x3. The limit from the left = ?

1 The limit from the right = ?

4 Are they equal?

NO Does the limit exist?

NO!!!

ONE-SIDED LIMITS:• As we approach 3 from the left, we are approaching

what value for the function? • What about from the right?

EVALUATING ONE-SIDED LIMITS:Consider the following problem:

When we do direct substitution, what do we get?

We can do some algebra: First factor…

Now calculate the limit:

2

1

2 3lim1x

x xx

2 3 12 3 31 1

x xx x xx x

22

1

1 2 1 32 3 0lim1 1 1 0x

x xx

2

1 1

2 3lim lim 3 ???1x x

x x xx

ANOTHER APPROACH:

x -1 -.999 -.99 -.9f(x) ?

What does it appear to be heading towards?

Did we get the same value as the previous technique?

2

1

2 3lim1x

x xx

APPLICATIONS Continuity

This is covered in the next section Asymptotes

This is used in Curve Sketching (without a calculator!)

HOMEWORK: Pg 67 #6-40E