Limits of Infinity
Transcript of Limits of Infinity
-
7/30/2019 Limits of Infinity
1/6
Limits at Infinity
Here we consider limits at infinity using Maple. Recall that the
definition of limx/N
f x = L is that giveneO 0, there is an
MO 0 such that if x > M, then f x L ! e. Graphically,
this means that if x is large enough, the graph off x lies in
the horizontal strip L e, LC e . See the following graph:
x
10 20 30 40
y
0.8
1.0
1.2
1.4
1.6
1.8
2.0
We see that for x bigger than, say, 19, the graph lies entirely in
the blue strip. Moreover, the oscillations seem to (and really
do) dampen out and the graph gets arbitrarily close toy= 1.
We can investigate limits at infinity with the computer or
-
7/30/2019 Limits of Infinity
2/6
calculator by (1) graphing the function for large x and
inspecting the graph to make a conjecture about the limit, or
(2) using built-in commands to calculate the limit. Of course,
you also need to be able to calculate limits at infinity without
the use of technology.
For the graphical approach, we consider the following
example.
Example 3
For each of the following functions:(i) Plot each function for large vlaues of x.
(ii) Use the plot to conjecture the limitof the function as x goes
toN
(iii) Determine precisely the limit of the function as x goes to
N.
a) e t = 1C t sin tt
b) f x =sin t
t
c)g x =2 x
3C 7 x
x2
Solution:
a) The plot of the first function for large values of t is:
-
7/30/2019 Limits of Infinity
3/6
x10 20 30 40 50
1
0.5
0
0.5
1
1.5
Clearly (why?) this function does not approach a horizontalline as x increases without bound. Therefore we do not expect
that there is a limit. (And there is none.)Question: Can you
carefully verify that there is no limit?
b) The plot of the function in part b for large values of t is:
-
7/30/2019 Limits of Infinity
4/6
t
50 100 150
0.4
0.2
0
0.2
0.4
0.6
0.8
We expect that the limit of this function is zero as t increases
without bound. Sincesin t
t%
1
t, and since
limt/N
1
t= 0 (why?), we have, via the sandwich theorem,
limt/N
sin t
t = 0.
c) The plot of the function is, for large x,
-
7/30/2019 Limits of Infinity
5/6
x
5 10 15 20 25 30
10
20
30
40
50
60
It seems clear that there is no horizontal asymptote here. In
fact, we may write2$x
3C 7$x
x2
=2 x
2C 7
x= 2 xC
7
xand
from the last expression it is easy to see that because2xgrows
without bound as x increases and because7
xgets arbitrarily
small as x increases without bound, the function grows like2x;
i.e. it does not have a limit.
We can calculate limits without appealing to a graph as well.
There is a built-in command "limit" that will calculate just
-
7/30/2019 Limits of Infinity
6/6
(2)(2)
OO
OO
(1)(1)
about every limit we're interested in. (Though it sometimes has
trouble...) For example, let's calculate limx/N
23$xCx
9xCx
3. All we
need to do is
limit2
3$xCx
9xCx
3, x=N ;
0
As another example we calculate limx/N 1C
5
x
2$x
:
limit 1C5
x
2$x
, x =N ;
e10