Limits at Infinity Lesson 4.5 What Happens? • We wish to investigate what happens when functions go … To infinity and beyond …
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Transcript Limits at Infinity Lesson 4.5 What Happens? • We wish to investigate what happens when functions go … To infinity and beyond …
Limits at Infinity
Lesson 4.5
What Happens?
• We wish to investigate what happens when
functions go …
To infinity and
beyond …
Limits with Infinity
• What happens to a function in the long run
lim f ( x) L 0, N1
x
such that f ( x) L whenever x N1
N1
L
Rules for Manipulating Limits
• Note rules on page 239
• Note special limits
c
lim r 0
x x
r
x
lim x 0
x e
r is a positive rational
number
Manipulating, Evaluating
• Symbolically
2
2
x 1
1
x2
1
x
x
lim
lim
lim
x 3 x 5
x
x
5 3
5
3
x3
x
x
• Use Calculator
limit((x+2)/(3x-5),x,+)
• Graph and observe
go to zero
Rational Functions
an x n ...
m
bm x ...
• Leading terms dominate
m = n => limit = an/bm
m > n => limit = 0
m < n => asymptote linear diagonal
or higher power polynomial
Rational Functions
• Vertical asymptotes
where denominator = 0
• Y-intercepts
where x = 0
• X-intercepts
where numerator = 0
Example
2 x 5x 3
f ( x)
2
x 9
• Find
2
horizontal asymptote
vertical asymptote(s)
zeros
y-intercept
Example
x 2x 1
f ( x)
3x 2
2
• Find
horizontal asymptote
vertical asymptote(s)
zeros
y-intercept
Limits Involving Trig Functions
• Consider f(x) = sin x
As x gets very large, function oscillates between
1 and -1
Thus no limit
• Consider
sin x
lim
x
x
Squeeze theorem applies
Limit is 0
1
x
Assignment
• Lesson 4.5
• Page 245
• Exercises 1 – 57 EOO
Also 99, 102