Warm up

Evaluate the limit

2

211. lim

3 4

x

x x

x x 0

1 12. lim

h

h

h

2

21

63. lim

2x

x x

x

0

4. lim25 5 x

x

x

Section 2.4

Continuity• SWBAT

– Define continuity and its types

Conceptual continuity

2.4 Continuity

• This implies :1. f(a) is defined2. f(x) has a limit as x approaches a3. This limit is actually equal to f(a) .

Definition (cont’d)

Types of discontinuity

Removable Discontinuity: “A hole in the graph”

(You can algebraically REMOVE the discontinuity)

Types of discontinuity (cont’d)

Infinite discontinuity:• Where the graph

approaches an asymptote

• It can not be algebraically removed

jump discontinuity the function “jumps” from one value to another.

Example• Where are each of the following

functions discontinuous, and describe the type of discontinuity

2

31.

12x

f xx x

2 9 202.

4x x

f xx

One-Sided Continuity• Continuity can occur from just one

side:

Continuity on an Interval

• So far continuity has been defined to occur (or not) one point at a time.

• We can also consider continuity over an entire interval at a time:

• Continuous on an Interval: it is continuous at every point on that interval.

Polynomials and Rational Functions

• Write the interval where this function is continuous.3 2

2

2 1lim :

5 3x

x xx

5 5( , ) ( , )

3 3

Types of Continuous Function

• We can prove the following theorem:

• This means that most of the functions encountered in calculus are continuous wherever defined.

1. Lim f(x)x2-

2. Lim f(x)x2+

3. Lim f(x)x-

4. Lim f(x)x-2-

5. Lim f(x) x-2+

6. Lim f(x)x0

7. f(2) 8. f(-2)

Assignment 8

• p. 126 1-31 odd

• Quiz tomorrow – 2.1 through 2.4 Continuity