Post on 26-Mar-2015
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.