Graphically What is happening in the graph below Graphically We can make the following statements ALSO Vertical Asymptotes When do vertical asymptotes occur algebraically Denominator 0 ID: 138197
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Slide1
2.2
Limits Involving InfinitySlide2
Graphically
What is happening in the graph below?Slide3
Graphically
We can make the following statements:
ALSO:Slide4
Vertical Asymptotes
When do vertical asymptotes occur algebraically?
Denominator
= 0
(a function is undefined…this includes trig functions)
Using Limits:
A vertical asymptote of
x = a
exists for a function if
ORSlide5
Horizontal Asymptotes
A horizontal asymptote of
y = b
exists if
OR
Example:
Identify all horizontal and vertical asymptotes of Slide6
Special Limits
Example:
What is
If we substitute in ∞, sin ∞ oscillates between -1 and 1, so we must find another way to show this limit algebraically.
USING SANDWICH THEOREM:Slide7
Special Limits
0
0
Therefore, by the Sandwich Theorem, Slide8
Special Limits
Example:
What isSlide9
Special Limits
Example:
What isSlide10
Limits Involving ±∞
The same properties of adding, subtracting, multiplying, dividing, constant multiplying, and using powers for limit also apply to limits involving infinity. (see pg. 71)Slide11
End Behavior
We sometimes want to how the ends of functions are behaving.
We can use much simpler functions to discuss end behavior than a complicated one that may be given.
To look at end behavior, we must use limits involving infinity.Slide12
End Behavior
A function
g
is an end behavior model for f if and only if
Right-end
behavior model when
x
+∞
Left-end behavior model when
x
-∞Slide13
End Behavior
Show that
g(x) = 3x
4 is an end behavior model for f(x) = 3x
4 – 2x
3 + 3x2 – 5x + 6.Slide14
Finding End Behavior Models
Find a right end behavior model for the function
f(x) = x +
ex
Notice when
x
is ∞,
e
∞ goes to 0.
If we use a function of
g(x) = x
in the denominator, we get
0
Therefore,
g(x) = x
is a right hand behavior model for
f(x)Slide15
Finding End Behavior Models
Find a left end behavior model for the function
f(x) = x +
ex
Notice when
x
is
∞, ex goes to
∞ and
x
goes to –∞
.
Which one has more effect on the left-end of the function? (Which one gets to ∞ faster?)
e
∞
Therefore, use e
–x
as a left-end behavior model for
f(x)
.Slide16
Finding End Behavior Models
Find a left end behavior model for the function
f(x) = x +
ex
0
1
Therefore, e
–x
is a left-end behavior model for
f(x)
.Slide17
HW
Section 2.2 (#1-7 odd, , 21, 23, 25, 27-33 odd, 39, 41, 43, 45-49 odd)
Web Assign due Monday night