2.2  The Limit of a Function

 

Conceptual idea of limit as "close to"

 

Intuitive Definition                       

  read as “the limit of f(x), as x approaches a, equals L”

If we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a) but not equal to a.

 

For , find the limit of f(x) as x ®4

 

I’ll demonstrate the concept using TABLE features          

Start 3.7  Dx = 0.1 AUTO

x

3.7

3.8

3.9

4.0

4.1

4.2

4.3

f(x)

 

 

 

 

 

 

 

 

Start 3.97  Dx = 0.01 AUTO

x

3.97

3.98

3.99

4.00

4.01

4.02

4.03

f(x)

 

 

 

 

 

 

 

 

Start 3.997  Dx = 0.001 AUTO  etc.

x

3.997

3. 998

3. 999

4.000

4. 001

4. 002

4. 003

f(x)

 

 

 

 

 

 

 

 

It appears that f(x) approaches 3.2 as x approaches 4, but f(4) ¹ 3.2

 

,  means that as x get "closer to" 4, f(x) get "closer to" 3.2

 

We could make f(x) as close to 3.2 as wish by choosing x "close enough"

 

Symbolically 

 

This limit could also be found algebraically.

 

Graphically:

[image] [image] [image]

                                     

 

Consider a function, which does not simplify algebraically

            Does it exist?  What does it equal?

Let's use TABLE and ASK; letting x approach 0 from left and right

Start -.3  Dx = 0.1 AUTO

x

-.3

-.2

-.1

0

.1

.2

.3

f(x)

 

 

 

 

 

 

 

 

Start -.03  Dx = 0.01 AUTO

x

-.03

-.02

-.01

0

.01

.02

.03

f(x)

 

 

 

 

 

 

 

 

Start -.003  Dx = 0.001 AUTO

x

-.003

-.002

-.001

0

.001

.002

.003

f(x)

 

 

 

 

 

 

 

It appears to approach 1, but not sure what happens to f(x) when closer than ±0.001

Look at Graph

[image]

Often the limit does not exist … for various reasons

          Look at the text examples on own.

Consider               

[image]                           

Consider 

[image]

Left and Right-hand Limits 

 

Intuitive Definition for Left Hand Limit                 

  read as “the limit of f(x), as x approaches a from the left, equals L”

If we can make the values of f(x) arbitrarily close to L  by taking x to be sufficiently close to a and x less than a..

Right hand Limit definition is similar    

Find limits as

a) 

 

 

b)   


Definition

         

 

Example – Transparency  - Problem #6

 

 

 

 

 

 

 

Infinite Limits

     

 

    

 

    

 

Definition - Vertical Asymptote

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

                                     

 

                                

 

 

 

Assignment: 

Pg 74; 1, 2, 5, 7, 9, 12, 13, 17, 25, 29, 31