2.7 Limits involving infinity
Thurs Oct 1
Do Now
Find
Undefined limits
• Let us examine
• Recall that it is undefined at x = 0, so
we say that it does not exist.
• We know there’s an asymptote
• If asymptote =
•From the left side, f(x) decreases without
bound. We can say that
•From the right side, f(x) increases without
bound. We can say that
One-sided Limits that agree
• If both one-sided limits agree, we can
say that the whole limit exists. This is
now true for infinity.
• Ex:
1
lim 2 = +¥
x®0 x
Ex 2
You try
Find the limits
1 - 2x
lim+ 2
x ®1 x - 1
4- x
lim+
2
x ® 2 ( x - 2)
Determining the sign of infinity
• To determine whether it is positive or
negative infinity, plug a # very close. We write
a + or - above/below each factor.
• Example:
Limits Approaching Infinity
• We are also interested in examining the
behavior of functions as x increases or
decreases without bound ( x ® ±¥)
1
lim = 0
x ®¥ x
1
lim = 0
x ® -¥ x
Thm- For any rational number t > 0,
1
lim t = 0
x ® ±¥ x
as long as there isn’t a negative in a radical
Thm- For any polynomial of degree n > 0, and
p(x) = an x n + an -1x n -1 + ...+ a0 then
Limits to infinity with a rational
function
• When the limit is approaching infinity and the function
is a fraction of two polynomials, we have to look at
the power of the numerator and denominator (The
highest exponent in each).
• We divide every term by the variable with the highest
power
• There are 3 cases
Limits to infinity with a rational
function pt2
• Case 1
– The power of the numerator is bigger
In this case, the limit = +/- infinity
Example: Evaluate
Limits to infinity with a rational
function pt3
• Case 2:
– The power of the denominator is bigger
In this case, the limit = 0
Example: Evaluate
Limits to infinity with a rational
function pt4
• Case 3:
– The powers are the same
In this case, we must look at the coefficients of the first
terms. The limit = the fraction of coefficients
Example: Evaluate
You Try
Evaluate each limit
1)
2)
3)
Closure
• What happens when we have a rational
function and the limit approaches
infinity?
• Describe the 3 cases
• HW p. 105 #7-15, 23-27 odds
2.7 Limits to infinity
Fri Oct 2
• Do Now
• Find each limit
- 3x + 2 x - 4
lim
x ®¥ 5 x 3 + x 2 + 1
3
lim
x ®¥
-x
4+ x
2
HW Review:p105 #7-15 23-27
•
•
•
•
•
•
•
•
7) 1
9) 0
11) 7/4
13) - infinity
15) + infinity
23) 0
25) 2
27) 1/16
More book ex if necessary
Limits approaching infinity
• What about rational functions that
aren’t only polynomials?
• Trig functions: they typically do not
exist.
• Ex: sin x
Exponential functions
• Notice exponential functions increase
much faster than algebraic polynomials.
Consider their ‘exponents’ as greater
than any polynomial’s exponent.
• Ex:
e^x increases faster, so the limit = 0
Logarithmic Functions
• The inverse of the previous slide is also
true.
• Logarithmic functions increase very
slowly as x approaches infinity
• Therefore, consider their “exponents”
as less than any algebraic polynomial.
• Ex:
x
lim
x ®¥
ln x
=¥
Classwork
• (green book) Worksheet p.122 #31-40
Closure
• Journal Entry: What did we learn about
functions with numerators and
denominators that are not polynomials?
• How do we determine those limits as x
approaches infinity?
• HW: Finish worksheet p. 122 #31-40
2.7 Limits Involving Infinity
Mon Oct 5
Do Now
Evaluate each limit
1)
2)
3)
HW Review: worksheet p.122
#31-40
31) infinity
36) 0
32) 0
37) infinity
33) DNE
38) neg inf
34) DNE
39) 0
35) 0
40) infinity
Practice
Worksheet 5-28
Closure
Hand in: Find the limits, if they exist
1)
2)
HW: Finish worksheet p.131-132 #7-28
2.7 Review
Wed Oct 2
•
•
•
•
•
•
•
•
•
•
•
HW Review worksheet p.13113218)#7-28
7) neg infinity
3/2
3
8) infinity
9) infinity
10) infinity
11) 3/2
12) 5/2
13) 0
14) 0
15) 0
16) 5/3
17) -3 5 /8
19) - 5
20) 5
6
1
21) 6 = 6
6
22) - 16 = - 6
23) 3
24) 3
25) neg infinity
26) infinity
27) -1/7
28) 4/7