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. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
Formal Definition of the Limit
According to the theorems we’ve seen thus far,
So, we should be able to make (3x + 4) as close as we like to 10, just by making x sufficiently close to 2.
But can we actually do this? For instance, can we force (3x + 4) to be within distance 1 of 10?
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4. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
Formal Definition of the Limit
To see what values of x will guarantee this, write an inequality that says that (3x + 4) is within 1 unit of 10: or
Solve for x:
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5. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
Formal Definition of the Limit
If x is between 2 – 1/3 and 2 + 1/3, 3x + 4 is guaranteed to be between 9 and 11.
While we can make (3x + 4) within
1 unit of 10, we wish to make it
arbitrarily close, as close as
anyone would ever demand.
We accomplish this by repeating
the arguments in this example , this time for an unspecified distance, call it ε (epsilon, where ε > 0).
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6. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.2
Verifying a Limit
Show that we can make (3x + 4) within any specified distance ε > 0 of 10 (no matter how small ε is), just by making x sufficiently close to 2.
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7. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.2
Verifying a Limit
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8. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.2
Verifying a Limit
No matter how close we are asked to make (3x + 4) to 10, we can accomplish this simply by taking x to be in the specified interval.
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9. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.3
Proving That a Limit Is Correct
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10. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.3
Proving That a Limit Is Correct
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11. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.3
Proving That a Limit Is Correct
Since x = 1,
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12. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.3
Proving That a Limit Is Correct
Taking δ = ε/2 and working backward, we see that requiring x to satisfy
will guarantee that
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13. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.1
(Precise Definition of Limit)
For a function f defined in some open interval containing a (but not necessarily at a itself), we say
if given any number ε > 0, there is another number δ > 0, such that 0 < |x − a| < δ guarantees that | f (x) − L| < ε.
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14. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.1
It is rather difficult to explicitly find δ as a function of ε, for all but a few simple examples.
Despite this, learning how to work through the definition, even for a small number of problems, will shed considerable light on a deep concept.
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15. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.4
Using the Precise Definition of Limit
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16. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.4
Using the Precise Definition of Limit
If this limit is correct, then given any ε > 0, there must be a δ > 0 for which 0 < |x − 2| < δ guarantees that
Notice that
Assume that x lies in the interval [1, 3]; thus:
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17. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.4
Using the Precise Definition of Limit
Finally, if we require that
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18. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.4
Using the Precise Definition of Limit
We now have two restrictions:
So, choose:
Working backward:
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19. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
Exploring the Definition of Limit Graphically
Finding a δ for a given ε is not always easily accomplished.
However, we can explore the definition graphically for any function.
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20. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.5
Exploring the Precise Definition of Limit Graphically
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21. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.5
Exploring the Precise Definition of Limit Graphically
In example 6.4, we saw that ,
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22. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.5
Exploring the Precise Definition of Limit Graphically
Confirmed.
We can draw virtually the same
picture for any given value of ε, since we have an explicit formula for finding δ given ε.
We are not proving that the above limit is correct. To prove this requires symbolically finding a δ for every ε > 0.
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23. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.7
Exploring the Definition of Limit
Where the Limit Does Not Exist
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24. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.7
Exploring the Definition of Limit
Where the Limit Does Not Exist
Choose
Find a δ > 0 for which 0 < |x| < δ guarantees that
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25. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.7
Exploring the Definition of Limit
Where the Limit Does Not Exist
Try δ = 0.1. Setting the x-range to
the interval [−0.1, 0.1] and the
y-range to the interval [0.5, 1.5]
and redrawing gives:
No points are plotted in the window for any x < 0.
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26. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.7
Exploring the Definition of Limit
Where the Limit Does Not Exist
In fact, there is no choice of δ that makes the defining inequality true for ε = 1/2.
Thus, the conjectured limit of 1 is incorrect.
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27. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
Limits Involving Infinity
Recall that we write
whenever the function increases without bound as x → a.
That is, we can make f (x) as large as we like, simply by making x sufficiently close to a.
So, given any large positive number, M, we must be able to make f (x) > M, for x sufficiently close to a.
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28. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.2
For a function f defined in some open interval containing a (but not necessarily at a itself), we say
if given any number M > 0, there is another number δ > 0, such that 0 < |x − a| < δ guarantees that f (x) > M.
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29. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.3
For a function f defined in some open interval containing a (but not necessarily at a itself), we say
if given any number N < 0, there is another number δ > 0, such that 0 < |x − a| < δ guarantees that f (x) < N.
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30. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.8
Using the Definition of Limit Where the Limit Is Infinite
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31. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.8
Using the Definition of Limit Where the Limit Is Infinite
Given any (large) number M > 0, we need to find a distance δ > 0 such that if x is within δ of 0 (but not equal to 0) then
Since both M and x2 are positive, this is equivalent to
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32. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.8
Using the Definition of Limit Where the Limit Is Infinite
So, for any M > 0, if we take and work
backward, we have that 0 < |x − 0| < δ guarantees that
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33. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
Limits Involving Infinity
If we write
we mean that as x increases without bound, f (x) gets
closer and closer to L.
That is, we can make f (x) as close to L as we like, by choosing x sufficiently large.
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34. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.4
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35. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.5
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36. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.9
Using the Definition of Limit Where x Tends to −∞
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Slide 36

37. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.9
Using the Definition of Limit Where x Tends to −∞
Since x < 0, |x| = −x. Thus:
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38. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.9
Using the Definition of Limit Where x Tends to −∞
Dividing both sides by ε and multiplying by x (remember that x < 0 and ε > 0, so that this will change the direction of the inequality), we get
So, if we take and work backward, we have satisfied the definition and thereby proved that the limit is correct.
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39. FORMAL DEFINITION OF THE LIMIT
1.6
FORMAL DEFINITION OF THE LIMIT
6.1
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Slide 39