2. Limits and Continuity 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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3. The Limit: Informal Idea
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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4. The Limit: Informal Idea
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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5. The Limit: Informal Idea
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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6. 1.2 THE CONCEPT OF LIMIT 2.1 Evaluating a Limit Slide 6
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7. 1.2 THE CONCEPT OF LIMIT 2.1 Evaluating a Limit Slide 7
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8. 1.2 THE CONCEPT OF LIMIT 2.1 Evaluating a Limit Slide 9
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9. 1.2 THE CONCEPT OF LIMIT 2.1 Evaluating a Limit
Since the two one-sided limits
of f (x) are the same, we summarize our results by saying that
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10. 1.2 THE CONCEPT OF LIMIT 2.1 Evaluating a Limit
We can also determine the limit algebraically.
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11. 1.2 THE CONCEPT OF LIMIT 2.2 A Limit that Does Not Exist Slide 15
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12. 1.2 THE CONCEPT OF LIMIT 2.2 A Limit that Does Not Exist Slide 16
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13. 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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14. 1.2 THE CONCEPT OF LIMIT 2.3 Determining Limits Graphically
Use the graph to determine
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15. 1.2 THE CONCEPT OF LIMIT 2.3 Determining Limits Graphically Slide 19
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16. 1.2 THE CONCEPT OF LIMIT 2.4 A Limit Where Two Factors Cancel Slide 16
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17. 1.2 THE CONCEPT OF LIMIT 2.4 A Limit Where Two Factors Cancel
From the left:
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18. 1.2 THE CONCEPT OF LIMIT 2.4 A Limit Where Two Factors Cancel
From the right:
Conjecture:
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19. 1.2 THE CONCEPT OF LIMIT 2.4 A Limit Where Two Factors Cancel Slide 23
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20. 1.2 THE CONCEPT OF LIMIT 2.4 A Limit Where Two Factors Cancel
Algebraic cancellation:
Likewise:
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21. 1.2 THE CONCEPT OF LIMIT 2.5 A Limit That Does Not Exist Slide 21
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22. 1.2 THE CONCEPT OF LIMIT 2.5 A Limit That Does Not Exist
From the right:
Conjecture:
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23. 1.2 THE CONCEPT OF LIMIT 2.5 A Limit That Does Not Exist
From the left:
Conjecture:
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24. 1.2 THE CONCEPT OF LIMIT 2.5 A Limit That Does Not Exist Conjecture:
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25. 1.2
THE CONCEPT OF LIMIT
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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