outline 1. motivation 2. limit of a function 3. one-sided...
TRANSCRIPT
Le c t u r e 1 | 1
Chapter 1
Limits and Continuity
Outline 1. Motivation
2. Limit of a function 3. One-sided limits 4. Infinite limits
Le c t u r e 1 | 2
1. Motivation: Tangent line problem
EX Find the slope of the tangent line to the curve
in the -plane at the point .
Le c t u r e 1 | 3
Le c t u r e 1 | 4
tends to 2 as gets closer to 1.
The tangent line at should have slope 2.
Le c t u r e 1 | 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.
Le c t u r e 1 | 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.
Le c t u r e 1 | 7
Le c t u r e 1 | 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
Le c t u r e 1 | 9
Remark We don’t mind the value of when in considering the limit. It may even not exist!
Le c t u r e 1 | 10
EX Guess the value of
using a calculator.
So
Le c t u r e 1 | 11
EX Estimate the value of
Thus we guess that
What if is taken further down to ?
Le c t u r e 1 | 12
A machine error!
Le c t u r e 1 | 13
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
Le c t u r e 1 | 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
Le c t u r e 1 | 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
Le c t u r e 1 | 16
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
Le c t u r e 1 | 17
Le c t u r e 1 | 18
Rule We have
if and only if
Le c t u r e 1 | 19
EX Investigate the limits of
as approaches 0.
Le c t u r e 1 | 20
EX From the graph of , find
Le c t u r e 1 | 21
4. Infinite Limits EX Find
if it exists.
We have
can be arbitrarily large by
taking close enough to 0.
Le c t u r e 1 | 22
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
Le c t u r e 1 | 23
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
Le c t u r e 1 | 24
can be defined in a similar fashion.
Le c t u r e 1 | 25
Le c t u r e 1 | 26
Rule We have if and only if
and
Similarly, if and only if
and
Le c t u r e 1 | 27
Def The line is called a vertical asymptote of the curve if at least one of the following statements is true:
Le c t u r e 1 | 28
EX Use the graph to find
Le c t u r e 1 | 29
EX Use the graph to find the vertical asymptotes of .