2.2 Limits Involving Infinity
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Transcript 2.2 Limits Involving Infinity
2.2
Limits Involving Infinity
Graphically
What is happening in the graph below?
Graphically
We can make the following statements:
lim f ( x )
x 0
lim f ( x )
x 0
ALSO:
lim
x
lim f ( x ) DNE
x 0
f ( x) 0
lim f ( x ) 0
x
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
lim f ( x )
x a
OR
lim f ( x )
x a
Horizontal Asymptotes
A horizontal asymptote of y = b exists if
lim
x
f ( x) b
OR lim f ( x ) b
x
Example:
Identify all horizontal and vertical asymptotes of
f ( x)
x
x 1
2
Special Limits
Example:
sin x
What is lim
x
x
If we substitute in ∞, sin ∞ oscillates between -1 and 1, so
we must find another way to show this limit algebraically.
USING SANDWICH THEOREM:
1 sin x 1
1
x
sin x
x
1
x
Special Limits
0
lim
x
1
0
sin x
lim
x
x
0 lim
x
x
sin x
x
lim
0
x
Therefore, by the Sandwich Theorem,
lim
x
sin x
x
0
1
x
Special Limits
Example:
sin x
What is lim
x 0
x
lim
x 0
sin x
x
1
Special Limits
Example:
cos x 1
What is lim
x 0
lim
x 0
cos x 1
x
x
0
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)
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.
End Behavior
A function g is an end behavior model for f
if and only if
lim
x
f ( x)
1
g ( x)
Right-end behavior model when x +∞
Left-end behavior model when x -∞
End Behavior
Show that g(x) = 3x4 is an end behavior
model for
f(x) = 3x4 – 2x3 + 3x2 – 5x + 6.
3x 2 x 3x 5x 6
4
lim
3
x
2
3x
4
2
1
5
2
lim 1
2
4
3
x
3x x
3x
x
1
Finding End Behavior Models
Find a right end behavior model for the
function f(x) = x + ex
xe
lim
x
?
x
1
Notice when x is ∞,
e∞ goes to 0.
If we use a function of g(x) = x in the denominator,
0
we get
lim
x
xe
x
x
x
x e
1
lim 1
lim
x
x x
x
x
xe
Therefore, g(x) = x is a right hand
behavior model for f(x)
1
Finding End Behavior Models
Find a left end behavior model for the
function f(x) = x + ex
lim
xe
x
?
x
1
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?)
Therefore, use e–x as a left-end behavior model
for f(x).
e∞
Finding End Behavior Models
Find a left end behavior model for the
function f(x) = x + ex
lim
x
lim
x
lim
x
xe
x
e
1
?
x
e
x
xe
x
xe
x
1
0
x
x
e
lim x x
x e
e
1
1
Therefore, e–x is a left-end behavior
model for f(x).
HW
Section 2.2 (#1-7 odd, , 21, 23, 25, 27-33
odd, 39, 41, 43, 45-49 odd)
Web Assign due Monday night