149
SECTION 1.6
.
.
Formal Definition of the Limit
121
What is the significance of the two vertical asymptotes? Ex
plain in physical terms what type of shot corresponds to each
vertical asymptote. Estimate the minimum value of
v
0
(call it
v
min
). Explain why it is easier to shoot a ball with a small ini
tial velocity. There is another advantage to this initial velocity.
Assume that the basket is 2 ft in diameter and the ball is 1
ft in diameter. For a free throw,
L
=
15 ft is perfect. What is
the maximum horizontal distance the ball could travel and still
go in the basket (without bouncing off the backboard)? What
is the minimum horizontal distance? Call these numbers
L
max
and
L
min
. Find the angle
θ
1
corresponding to
v
min
and
L
min
and
the angle
θ
2
corresponding to
v
min
and
L
max
. The difference

θ
2
−
θ
1

is the angular margin of error. Brancazio has shown
that the angular margin of error for
v
min
is larger than for any
other initial velocity.
2.
In applications, it is common to compute
lim
x
→∞
f
(
x
) to de
termine the “stability” of the function
f
(
x
). Consider the
function
f
(
x
)
=
xe
−
x
. As
x
→ ∞
, the first factor in
f
(
x
)
goes to
∞
, but the second factor goes to 0. What does the
product do when one term is getting smaller and the other
term is getting larger? It depends on which one is chang
ing faster. What we want to know is which term “domi
nates.” Use graphical and numerical evidence to conjecture
the value of lim
x
→∞
(
xe
−
x
). Which term dominates? In the limit
lim
x
→∞
(
x
2
e
−
x
), which term dominates? Also, try lim
x
→∞
(
x
5
e
−
x
).
Based on your investigation, is it always true that exponentials
dominate polynomials? Are you positive? Try to determine
which type of function, polynomials or logarithms, domi
nates.
1.6
FORMAL DEFINITION OF THE LIMIT
We have now spent many pages discussing various aspects of the computation of limits.
This may seem a bit odd, when you realize that we have never actually
defined
what a limit
is. Oh, sure, we have given you an
idea
of what a limit is, but that’s about all. Once again,
we have said that
lim
x
→
a
f
(
x
)
=
L
,
if
f
(
x
) gets closer and closer to
L
as
x
gets closer and closer to
a
.
So far, we have been quite happy with this somewhat vague, although intuitive, de
scription. In this section, however, we will make this more precise, and you will begin to
see how
mathematical analysis
(that branch of mathematics of which the calculus is the
most elementary study) works.
Studying more advanced mathematics without an understanding of the precise defi
nition of limit is somewhat akin to studying brain surgery without bothering with all that
background work in chemistry and biology. In medicine, it has only been through a careful
examination of the microscopic world that a deeper understanding of our own macroscopic
world has developed, and good surgeons need to understand what they are doing
and why
they are doing it. Likewise, in mathematical analysis, it is through an understanding of the
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 Spring '10
 Lagerstrom
 Topology, Limit, lim g

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