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Chapter 1
Limits and Continuity
Outline 1. Motivation
2. Limit of a function
3. One-sided limits
4. Infinite limits
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1. Motivation: Tangent line problem
EX Find the slope of the tangent line to
the curve
in the
-plane at the point
.
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tends to 2 as
The tangent line at
gets closer to 1.
should have slope 2.
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The velocity problem
EX Suppose that a ball is drop from a
tower 450 m above the ground. Find the
velocity of the ball shortly after 5 seconds.
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Distance dropped at time :
Average velocity during
to
:
So the velocity shortly after second
should be
m/s.
It is called the instantaneous velocity.
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2. Limit of a Function
Suppose we are given a function
and a number .
Def If there is a number such that
can be arbitrary close to by taking
(1)
(2)
sufficiently close to
and
,
then we say that
the limit of
and we write
, as approaches
, equals
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Remark We don’t mind the value of
when
in considering the limit. It
may even not exist!
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EX Guess the value of
using a calculator.
So
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EX Estimate the value of
Thus we guess that
What if is taken further down to ?
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A machine error!
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Def If the value of
does not close to
any number as approaches , we say
that
the limit of
or simply
as approaches
, does not exist
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EX Guess the value of
As approaches , the values of
oscillate between
and infinitely
often, so it does not approaches any fixed
number. Thus
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3. One-Sided Limits
Def If there is a number such that we
can make the values of
arbitrarily
close to by taking
(1)
sufficiently close to
(2)
and
,
then we say that
the left-hand limit of
as
approaches is equal to
and write
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Def If there is a number such that we
can make the values of
arbitrarily
close to by taking
(1)
sufficiently close to
(2)
and
, then we say that
then we say that
the right-hand limit of
as
approaches is equal to
and write
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Rule
We have
if and only if
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EX Investigate the limits of
as
approaches 0.
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EX From the graph of
, find
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4. Infinite Limits
EX Find
if it exists.
We have can be arbitrarily large by
taking close enough to 0.
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Def Let be a function defined on both
sides of , except possibly at itself.
If
can be arbitrarily large by taking
(1)
sufficiently close to
(2)
and
,
then we say that
the limit of
or
as
is
approaches
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Def Let be a function defined on both
sides of , except possibly at itself.
If
can be arbitrarily small by taking
(1)
sufficiently close to
(2)
and
,
then we say that
the limit of
or
is
as
approaches
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can be defined in a similar fashion.
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Rule
We have
if and only if
and
Similarly,
and
if and only if
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Def The line
is called a vertical
asymptote of the curve
if at
least one of the following statements is
true:
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EX Use the graph to find
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EX Use the graph to find the vertical
asymptotes of
.