Section 2.2a. Limits Involving Infinity We can say the limit of f as x approaches infinity, meaning...
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Transcript of Section 2.2a. Limits Involving Infinity We can say the limit of f as x approaches infinity, meaning...
![Page 1: Section 2.2a. Limits Involving Infinity We can say the limit of f as x approaches infinity, meaning the limit of f as x moves increasingly far to the.](https://reader036.fdocuments.net/reader036/viewer/2022081807/5a4d1b6d7f8b9ab0599b3ebe/html5/thumbnails/1.jpg)
LIMITS INVOLVING INFINITYSection 2.2a
![Page 2: Section 2.2a. Limits Involving Infinity We can say the limit of f as x approaches infinity, meaning the limit of f as x moves increasingly far to the.](https://reader036.fdocuments.net/reader036/viewer/2022081807/5a4d1b6d7f8b9ab0599b3ebe/html5/thumbnails/2.jpg)
Limits Involving Infinity
We can say “the limit of f as x approaches infinity,”meaning the limit of f as x moves increasingly far to theright on the number line, or…
Note: “Infinity” does not represent a real number……however,
Saying “the limit of f as x approaches negative infinity”means the limit of f as x moves increasingly far to theleft.
![Page 3: Section 2.2a. Limits Involving Infinity We can say the limit of f as x approaches infinity, meaning the limit of f as x moves increasingly far to the.](https://reader036.fdocuments.net/reader036/viewer/2022081807/5a4d1b6d7f8b9ab0599b3ebe/html5/thumbnails/3.jpg)
Limits Involving InfinityThe graph of the reciprocal function:
1f xx
Our new limits:
1lim 0x x
1lim 0
x x
The line y = 0 is a horizontalasymptote of the graph of f…
![Page 4: Section 2.2a. Limits Involving Infinity We can say the limit of f as x approaches infinity, meaning the limit of f as x moves increasingly far to the.](https://reader036.fdocuments.net/reader036/viewer/2022081807/5a4d1b6d7f8b9ab0599b3ebe/html5/thumbnails/4.jpg)
Definition: Horizontal AsymptoteThe line y = b is a horizontal asymptote of the graphof a function y = f(x) if either
or limx
f x b
limx
f x b
Ex: Find any H.A. of the graph of 2 1f x x
lim lim 2x x
f x f x
H.A.: y = 2
![Page 5: Section 2.2a. Limits Involving Infinity We can say the limit of f as x approaches infinity, meaning the limit of f as x moves increasingly far to the.](https://reader036.fdocuments.net/reader036/viewer/2022081807/5a4d1b6d7f8b9ab0599b3ebe/html5/thumbnails/5.jpg)
Definition: Horizontal AsymptoteThe line y = b is a horizontal asymptote of the graphof a function y = f(x) if either
or limx
f x b
limx
f x b
Ex: Find any H.A. of the graph of 2 1
xf xx
lim 1x
f x
H.A.: y = –1, y = 1
Investigate with both agraph and a table… lim 1
xf x
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Sandwich Theorem Revisited
Find limx
f x
for sin xf xx
First, what do the graph and table suggest???
1 sin 1x Confirm Analytically:
1 sin 1xx x x
For x > 0, we have
1 sin 10 lim lim lim 0x x x
xx x x
And by the Sandwich Theorem:
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Limits Involving InfinityProperties of Our New Limits:Note: All of the properties for limits approaching realnumbers also hold for limits approaching infinity!!!
Including Sum Rule, Difference Rule, Product Rule,Constant Multiple Rule, Quotient Rule, Power Rule
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Limits Involving Infinity
Find5 sinlim
x
x xx
Rewrite:
5 sinlimx
x xx x
sinlim5 limx x
xx
5 0 5
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Limits Involving InfinitySometimes, a function outgrows all bounds (eitherpositive or negative) as x approaches a finite numbera we write:
limx a
f x
or limx a
f x
Think back to the reciprocal function:
0
1limx x
and0
1limx x
The line x = 0 is a vertical asymptote of the graph of f…
![Page 10: Section 2.2a. Limits Involving Infinity We can say the limit of f as x approaches infinity, meaning the limit of f as x moves increasingly far to the.](https://reader036.fdocuments.net/reader036/viewer/2022081807/5a4d1b6d7f8b9ab0599b3ebe/html5/thumbnails/10.jpg)
Definition: Vertical AsymptoteThe line x = a is a vertical asymptote of the graphof a function y = f(x) if either
or limx a
f x
limx a
f x
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Limits Involving InfinityFor the given function, (a) find the vertical asymptotes; (b) describethe behavior of the function to the left and right of each V.A.
2
12 5 3
xf xx x
Now, check the graph!V.A.:1 , 32
x x
0.5limx
f x
0.5limx
f x
1
2 1 3x
x x
3
limx
f x
3
limx
f x