MA111 CALCULUS I
Friday, 2/10/12
Today:
Questions on hand-in HW #1?
Questions on exercises?
Finish slides from Thursday
Horizontal and vertical asymptotes
Reading:
Exercises:
2.6
2.6, p.140: 1, 9, 15, 19, 28, 31, 35, 39 and 43
Friday, 2/10/12, Slide #1
Horizontal and Vertical Asymptotes
The horizontal line y = L is a horizontal asymptote of
the function y = f(x) if either
The vertical line x = A is a vertical asymptote of the
function y = f(x) if either
limxÆ+• f(x)= L or
limxÆ-• f(x)= L
limxÆA f(x)= ±• or
limxÆA¯ f(x)= ±•
Example:
What are the horizontal and vertical asymptotes of
f(x) = 3 + 1/(x2 – 1)?
Try to sketch the graph and then compare with Mathematica.
Friday, 2/10/12, Slide #2
Infinite Limits: tan(x) and arctan(x)
The function tan(x) is not a
one-to-one function, so it
doesn’t have an inverse,
unless we restrict its domain
If we restrict tan(x) to the domain
-p/2 < x < p/2, it’s one-to-one on
its range -• < tan x < +•.
So define its inverse function tan-1x,
also denoted arctan x, with domain
-• < x < +• and range
-p/2 < arctan x < p/2:
For any x, y = arctan x means
tan y = x and -p/2 < y < p/2
What are the horizontal and vertical
asymptotes of tan x and arctan x?
Friday, 2/10/12, Slide #3
CQ #1
Evaluate: limxÆ+• sin(x)
A. Does not exist
B. 1
C. 0
D. -1
E. +¶
Friday, 2/10/12, Slide #4
CQ #2
Evaluate: limxÆ+• sin(x)
x
A. Does not exist
B. 1
C. 0
D. -1
E. +¶
Friday, 2/10/12, Slide #5
Some more limit examples
What is limx Ø ¶ 3x / (x 2 +5) ?
What is limx Ø ¶ 3x
2
/ (x 2 +5) ?
What is limx Ø ¶ 3x
3
/ (x 2 +5) ?
Friday, 2/10/12, Slide #6