L ecture1| 1 Chapter 1
Limits and Continuity
Outline 1. Motivation
2. Limit of a function 3. One-sided limits
4. Infinite limits
L ecture1| 2 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 gets closer to 1.
The tangent line at should have slope 2.
L ecture1| 5 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.
L ecture1| 6 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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L ecture1| 8 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) sufficiently close to and (2) ,
then we say that
the limit of , as approaches , equals
and we write
L ecture1| 9 Remark We don’t mind the value of when in considering the limit. It
may even not exist!
L ecture1| 10 EX Guess the value of
using a calculator.
So
L ecture1| 11 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 as approaches , does not exist
or simply
L ecture1| 14 EX Guess the value of
As approaches , the values of oscillate between and infinitely
often, so it does not approaches any fixed number. Thus
L ecture1| 15 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 and (2) ,
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 and (2) , 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
L ecture1| 19 EX Investigate the limits of
as approaches 0.
L ecture1| 20 EX From the graph of , find
L ecture1| 21 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 and
(2) ,
then we say that
the limit of as approaches is
or
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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 and
(2) ,
then we say that
the limit of as approaches is
or
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can be defined in a similar fashion.
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Rule
We have if and only if
and
Similarly, if and only if
and
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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:
L ecture1| 28 EX Use the graph to find
L ecture1| 29 EX Use the graph to find the vertical
asymptotes of .
