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.