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2.2 Limits Involving Infinity. As the denominator gets larger, the value of the fraction gets smaller. There is a horizontal asymptote if:. or. This number becomes insignificant as. There is a horizontal asymptote at 1. Example 1:. Find:. - PowerPoint PPT Presentation
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2.2 Limits Involving Infinity
1f x
x
1lim 0x x
As the denominator gets larger, the value of the fraction gets smaller.
There is a horizontal asymptote if:
limx
f x b
or limx
f x b
2lim
1x
x
x
Example 1:
2limx
x
x
This number becomes insignificant as .x
limx
x
x 1
There is a horizontal asymptote at 1.
sin xf x
x
Example 2:
sinlimx
x
x Find:
When we graph this function, the limit appears to be zero.1 sin 1x
so for :0x 1 sin 1x
x x x
1 sin 1lim lim limx x x
x
x x x
sin0 lim 0
x
x
x
by the sandwich theorem:
sinlim 0x
x
x
Example 3: 5 sinlimx
x x
x
Find:
5 sinlimx
x x
x x
sinlim 5 limx x
x
x
5 0
5
Infinite Limits:
1f x
x
0
1limx x
As the denominator approaches zero, the value of the fraction gets very large.
If the denominator is positive then the fraction is positive.
0
1limx x
If the denominator is negative then the fraction is negative.
vertical asymptote at x=0.
Example 4:
20
1limx x
20
1limx x
The denominator is positive in both cases, so the limit is the same.
20
1 limx x
Humm….
End Behavior Models:
End behavior models model the behavior of a function as x approaches infinity or negative infinity.
A function g is:
a right end behavior model for f if and only if
lim 1x
f x
g x
a left end behavior model for f if and only if
lim 1x
f x
g x
Test ofmodel
Our modelis correct.
xf x x e Example 5:
As , approaches zero.x xe(The x term dominates.)
g x x becomes a right-end behavior model.
limx
x
x e
x
lim1
x
x
e
x
1 0 1
xh x e becomes a left-end behavior model.
limx
xx
x e
e
lim 1xx
x
e 0 1 1
As , increases faster than x decreases,x xe
therefore is dominant.xe
Test ofmodel
Our modelis correct.
xf x x e Example 5:
g x x becomes a right-end behavior model.
xh x e becomes a left-end behavior model.
On your calculator, graph:
1
2
3
x
x
y x
y e
y x e
10 10x
1 9y
Use:
5 4 2
2
2 1
3 5 7
x x xf x
x x
Example 6:
Right-end behavior models give us:
5
2
2
3
x
x
32
3
x
dominant terms in numerator and denominator
f(x)limx
Example 7:
021
42
27
5
xx
x
27
25
34132
)(xx
xxxf
Right-end behavior models give us:
dominant terms in numerator and denominator
f(x)limx
Example 8:
21
42
5
5
xx
25
25
34132
)(xx
xxxf
Right-end behavior models give us:
dominant terms in numerator and denominator
f(x)limx
Often you can just “think through” limits.
1lim sinx x
0
0lim sinx
x
0
1lim sinx x
zsinlim0z
0
Definition of a Limit Let c and L be real numbers. The function f has limit L as x approaches c
(x≠c), if, given any positive Ɛ, there is a positive number ɗ such that for all x, if x is within ɗ units of c, then f(x) is within Ɛ units of L.
0 < |x - c|< ɗ such that |f(x) – L| < Ɛ
Then we write Lf(x)limcx
Shortened version: Lf(x)lim
cx
If and only if for any number Ɛ >0, there is a real number ɗ >0such that if x is within ɗ units of c (but x ≠ c), then f(x) is withinƐ units of L.
C=2
L=3
3+Ɛ
3-Ɛ
→ ← ɗ = 1/3 ɗ is as large as possible. The graph just fits within the horizontal lines.
2274
)(2
x
xxxf
1. Plot the graph of f(x). Use a friendly window that includes x = 2 as a grid point. Name the feature present at x = 2.2. From the graph, what is the limit of f(x) as x approaches 2.3. What happens if you substitute x = 2 into the function?4. Factor f(x). What is the value of f(2)?5. How close to 2 would you have to keep x in order for f(x) tobe between 8.9 and 9.1?6. How close to 2 would you have to keep x in order for f(x) tobe within 0.001 unit of the limit in part 2? Answer in the form“x must be within ____ units of 2”7. What are the values of L, c, Ɛ and ɗ?8. Explain how you could find a suitable ɗ no matter how small Ɛ is.9. What is the reason for the restriction “… but not equal to c” in the definition of a limit?
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