3.5 Limits at Infinity Objective: Determine (finite) limits at infinity

Miss Battaglia
AP Calculus AB/BC
x decreases without bound
x increases without bound
x
-∞
0
f(x)
3
-100
-10
-1
1
10
100
2.9997 2.97 1.5 0 1.5 2.97 2.9997
lim f (x) = 3
f(x) approaches 3
x®-¥
lim (2x + 5)
x®-3
∞
3
f(x) approaches 3
lim f (x) = 3
x®-¥
lim f (x) = 3
x®¥
Let L be a real number.
1.
2.
The statement lim f (x) = L means that for each
x®¥
ε>0 there exists an M>0 such that |f(x)-L|<ε
whenever x>M.
The statement lim f (x) = L means that for each
x®-¥
ε>0 there exists an N<0 such that |f(x)-L|<ε
whenever x<N.
The line y=L is a horizontal asymptote of the
graph of f if
lim f (x) = L
x®-¥
or
lim f (x) = L
x®¥
If r is a positive rational number and c is any
real number, then
c
lim r = 0
x®¥ x
Furthermore, if xr is defined when x<0, then
c
lim r = 0
x®-¥ x
Find the limit:
2
lim(5 - 2 )
x®¥
x
Find the limit:
2x -1
lim
x®¥ x +1
Find each limit.
2x + 5
a. lim 2
x®¥ 3x +1
2x + 5
b. lim 2
x®¥ 3x +1
2
2x + 5
c. lim 2
x®¥ 3x +1
3
1.
2.
3.
If the degree of the numerator is less than the
degree of the denominator, then the limit of
the rational function is 0.
If the degree of the numerator is equal to the
degree of the denominator, then the limit of
the rational function is the ratio of the leading
coefficients.
If the degree of the numerator is greater than
the degree of the denominator, then the limit
of the rational function does not exist.
Find each limit.
a. lim
x®¥
3x - 2
2x +1
2
b. lim
x®-¥
3x - 2
2x +1
2
Find each limit.
a.
lim sin x
x®¥
b.
sin x
lim
x®¥
x
Suppose that f(t) measures the level of oxygen in a pond,
where f(t)=1 is the normal (unpolluted) level and the time t
is measured in weeks. When t=0, organic waste is dumped
into the pond, and as the waste material oxidizes, the level
of oxygen in the pond is
t 2 - t +1
f (t) = 2
t +1
What percent of the normal level of oxygen exists in the
pond after 1 week? After 2 weeks? After 10 weeks? What is
the limit as t approaches infinity?
(1,.5) (2,.6)
(10,.9)
Let f be a function defined on the interval (a,∞)
1.
2.
The statement lim f (x) = ¥ means that for each
x®¥
positive number M, there is a corresponding
number N>0 such that f(x)>M whenever x>N.
The statement lim f (x) = -¥ means that for each
x®¥
negative number M, there is a corresponding
number N>0 such that f(x)<M whenever x>N.
Find each limit.
lim x
a. x®¥
3
b.
lim x
x®-¥
3
Find each limit.
2x - 4x
a. lim
x®¥
x +1
2
2x - 4x
lim
b. x®-¥
x +1
2
 AB/BC
Calculus: Read 3.5 Page 205
#1-6, 19-35 odd, Quiz tomorrow