Limits and Continuity
CHAPTER
1
1.1 A BRIEF PREVIEW OF CALCULUS: TANGENT LINES
AND THE LENGTH OF A CURVE
1.2 THE CONCEPT OF LIMIT
1.3 COMPUTATION OF LIMITS
1.4 CONTINUITY AND ITS CONSEQUENCES
1.5 LIMITS INVOLVING INFINITY; ASYMPTOTES
1.6 FORMAL DEFINITION OF THE LIMIT
1.7 LIMITS AND LOSS OF SIGNIFICANCE ERRORS
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1.2
THE CONCEPT OF LIMIT
The Limit: Informal Idea
In this section, we develop the notion of limit using some
common language and illustrate the idea with some
simple examples.
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1.2
THE CONCEPT OF LIMIT
The Limit: Informal Idea
Suppose a function f is defined for all x in an open
interval containing a, except possibly at x = a.
If we can make f (x) arbitrarily close to some number L
(i.e., as close as we’d like to make it) by making x
sufficiently close to a (but not equal to a), then we say
that L is the limit of f (x), as x approaches a, written
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1.2
THE CONCEPT OF LIMIT
The Limit: Informal Idea
For instance, we have
since as x gets closer and closer to 2, f (x) = x2 gets closer
and closer to 4.
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Slide 5
1.2
THE CONCEPT OF LIMIT
EXAMPLE 2.1
Evaluating a Limit
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1.2
THE CONCEPT OF LIMIT
EXAMPLE 2.1
Evaluating a Limit
Solution
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Slide 7
1.2
THE CONCEPT OF LIMIT
EXAMPLE 2.1
Evaluating a Limit
Solution
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1.2
THE CONCEPT OF LIMIT
EXAMPLE 2.1
Evaluating a Limit
Solution
Since the two one-sided limits
of f (x) are the same, we
summarize our results by saying
that
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1.2
THE CONCEPT OF LIMIT
EXAMPLE 2.1
Evaluating a Limit
Solution
We can also determine the limit algebraically.
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1.2
THE CONCEPT OF LIMIT
EXAMPLE 2.2
A Limit that Does Not Exist
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Slide 15
1.2
THE CONCEPT OF LIMIT
EXAMPLE 2.2
A Limit that Does Not Exist
Solution
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1.2
THE CONCEPT OF LIMIT
A limit exists if and only if both corresponding one-sided
limits exist and are equal. That is,
In other words, we say that
if we can make f (x) as close as we might like to
L, by making x sufficiently close to a (on either side of a),
but not equal to a.
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Slide 13
1.2
THE CONCEPT OF LIMIT
EXAMPLE 2.4
A Limit Where Two Factors Cancel
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1.2
THE CONCEPT OF LIMIT
EXAMPLE 2.4
A Limit Where Two Factors Cancel
Solution
From the left:
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1.2
THE CONCEPT OF LIMIT
EXAMPLE 2.4
A Limit Where Two Factors Cancel
Solution
From the right:
Conjecture:
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1.2
THE CONCEPT OF LIMIT
EXAMPLE 2.4
A Limit Where Two Factors Cancel
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Slide 23
1.2
THE CONCEPT OF LIMIT
EXAMPLE 2.4
A Limit Where Two Factors Cancel
Solution
Algebraic cancellation:
Likewise:
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1.2
THE CONCEPT OF LIMIT
EXAMPLE 2.5
A Limit That Does Not Exist
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Slide 19
1.2
THE CONCEPT OF LIMIT
EXAMPLE 2.5
A Limit That Does Not Exist
Solution
From the right:
Conjecture:
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1.2
THE CONCEPT OF LIMIT
EXAMPLE 2.5
A Limit That Does Not Exist
Solution
From the left:
Conjecture:
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1.2
THE CONCEPT OF LIMIT
EXAMPLE 2.5
A Limit That Does Not Exist
Solution
Conjecture:
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1.2
THE CONCEPT OF LIMIT
REMARK 2.1
Computer or calculator computation of limits is
unreliable.
We use graphs and tables of values only as (strong)
evidence pointing to what a plausible answer might be.
To be certain, we need to obtain careful verification of
our conjectures. We explore this in sections 1.3–1.7.
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Slide 23
1.2
THE CONCEPT OF LIMIT
REMARK 2.3
Determining Limits Graphically
Use the graph to determine
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1.2
THE CONCEPT OF LIMIT
REMARK 2.3
Determining Limits Graphically
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1.2
THE CONCEPT OF LIMIT
REMARK 2.3
Determining Limits Graphically
Use the graph to determine
lim f ( x)  ________
3
x 0 
y
2
1
lim f ( x)  ________
x
x 0
-5
lim f ( x)  ________
x 0
-4
-3
-2
-1
-1
1
2
3
4
5
-2
-3
f (0)  ________
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1.2
THE CONCEPT OF LIMIT
REMARK 2.3
Determining Limits Graphically
Use the graph to determine
lim f ( x)  ________
3
x 2 
y
2
lim f ( x)  ________
1
x 2 
lim f ( x)  ________
-5
-4
-3
-2
-1
-1
x
1
2
3
4
5
x 2
-2
f (2)  ________
-3
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1.2
THE CONCEPT OF LIMIT
REMARK 2.3
Determining Limits Graphically
Use the graph to determine
lim f ( x)  ________
x1
lim f ( x)  ________
x 1
lim f ( x)  ________
lim f ( x)  ________
x 1
f (1)  ________
lim f ( x)  ________
x 0 
x 0 
lim f ( x)  ________
f (0)  ________
x 0
lim  f ( x)  _____
x 2
lim f ( x)  _____
x 2
lim f ( x)  _____
x 2
f (2)  _____
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Slide 18
1.2
THE CONCEPT OF LIMIT
REMARK 2.3
Determining Limits Graphically
Use the graph to determine
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1.2
THE CONCEPT OF LIMIT
REMARK 2.3
Determining Limits Graphically
Use the graph to determine
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1.2
THE CONCEPT OF LIMIT
REMARK 2.3
Determining Limits Graphically
Use Numerical and geographical evidence to find the limit.
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Slide 18
1.2
THE CONCEPT OF LIMIT
REMARK 2.3
Determining Limits Graphically
Use Numerical and geographical evidence to find the limit.
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Slide 18
1.2
THE CONCEPT OF LIMIT
REMARK 2.3
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Slide 18
1.2
THE CONCEPT OF LIMIT
REMARK 2.3
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1.2
THE CONCEPT OF LIMIT
REMARK 2.3
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1.2
THE CONCEPT OF LIMIT
REMARK 2.3
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1.2
THE CONCEPT OF LIMIT
REMARK 2.3
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1.2
THE CONCEPT OF LIMIT
REMARK 2.3
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1.2
THE CONCEPT OF LIMIT
REMARK 2.3
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1.2
THE CONCEPT OF LIMIT
REMARK 2.3
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1.2
THE CONCEPT OF LIMIT
REMARK 2.3
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1.2
THE CONCEPT OF LIMIT
REMARK 2.3
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1.2
THE CONCEPT OF LIMIT
REMARK 2.3
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Slide 18