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
x2
1
x

x
lim
 lim
 lim

x  3 x  5
x  
x

5 3
5
3

x3 
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