Please correct your homework as efficiently as possible so that we have plenty of time to get through the lesson.

Please correct your homework as efficiently as possible so that we have plenty of time to get through the lesson.

Limits, Asymptotes, and Continuity

Ex.2

5lim 2 2x

x x

Ex.2

1lim

3x x

Ex.5 2lim 4 2 2

xx x

Ex.3lim 2 1

xx x

Ex.

4

4 2

5 1lim

4x

x x

x x x

Ex.

2

4

2lim

3 2x

x

x x

Ex.0

sinlimx

x

x

Ex. 25

2lim

25x

x

x

Ex.2

5

6 5lim

5x

x x

x

4

2

-2

-5 5

Def. A horizontal asymptote of f (x) occurs at y = L if or lim

xf x L

lim

xf x L

Def. A vertical asymptote of f (x) occurs at values of x where f (x) is undefined (sort of).

• and are

examples of graphs that have a hole.

sin xy

x

2 6 5

5

x xy

x

Ex. Find all asymptotes of , then sketch the graph.

2

3

xy

x

4

2

-2

-5 5

• means that x approaches 2 from the right (larger than 2)

• means that x approaches 2 from the left (smaller than 2)

One-Sided Limits

2limx

2limx

Ex. 3

2lim

3x

x

x

Thm.

• The limit exists if both sides agree.

lim if lim limx a x a x a

f x L f x L f x

4

2

-2

-5 5

f x = -0.3x-2 2+4

Ex. For the function given, find:

a.

b.

c.

2

limx

f x

2

limx

f x

2

limx

f x

Ex. Find if 1

limx

f x

5 2 1

2 1 1

x xf x

x x

Def. (loose) A function is continuous on an interval if the graph has no gaps, jumps, or breaks on the interval.

Ex. Is continuous on [0,5]? 1

2f x

x

Def. (tight) A function f (x) is continuous on an interval if, for all points c on the interval:

i. exists

ii. exists

iii.

limx c

f x

f c

limx c

f x f c

Ex. Let

Find a value of B so that f (x) is continuous at x = 0.

1

02

0

xex

f x xB x

The test on Chapter 1 will be on Monday.

Next class we will review and I’ll pass out a Sample Test so you know what types of questions I can ask.