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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 gets closer to 1.

The tangent line at 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) sufficiently close to and

(2) ,

then we say that

the limit of , as approaches , equals and we write

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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 as approaches , does not exist or simply

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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 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

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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 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:

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EX Use the graph to find

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EX Use the graph to find the vertical asymptotes of .