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. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
Continuity: Informal Idea
We say that a function is continuous on an interval if its graph on that interval can be drawn without interruption, that is, without lifting the pencil from the paper.
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4. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.1
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5. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.1
For f to be continuous at x = a, the definition says that
(i) f (a) must be defined,
(ii) the limit must exist
and
(iii) the limit and the value of f at the point must be the same.
Further, this says that a function is continuous at a point exactly when you can compute its limit at that point by simply substituting in.
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6. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
Examples of Points of Discontinuity
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7. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.1
Finding Where a Rational Function Is Continuous
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8. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.1
Finding Where a Rational Function Is Continuous
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9. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.2
Removing a Hole in the Graph
Extend the function from example 4.1 to make it continuous everywhere by redefining it at a single point.
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10. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.2
Removing a Hole in the Graph
Let
for some real number a. If we let a = 4,
Then
and g(x) is continuous at x = 1.
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11. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
Removable Discontinuities
When we can remove a discontinuity by redefining the function at that point, we call the discontinuity removable. Not all discontinuities are removable, however.
Removable
Not removable
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12. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.3
Functions That Cannot Be Extended Continuously
Show that (a) and (b) cannot
be extended to a function that is continuous everywhere.
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13. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.3
Functions That Cannot Be Extended Continuously
Hence, no matter how we might define f (0), f will not be continuous at x = 0.
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14. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.3
Functions That Cannot Be Extended Continuously
Due to the endless oscillation, the limit does not exist, and there is no way to redefine the function at x = 0 to make it continuous there.
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15. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.1
All polynomials are continuous everywhere.
Additionally, sin x and cos x are continuous everywhere.
is continuous for all x, when n is odd and for x > 0, when n is even.
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16. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.2
Suppose that f and g are continuous at x = a. Then all of the following are true:
Simply put, Theorem 4.2 says that a sum, difference or product of continuous functions is continuous, while the quotient of two continuous functions is continuous at any point at which the denominator is nonzero.
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17. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.3
Suppose that and f is continuous at L. Then,
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18. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.1
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19. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.5
Continuity for a Composite Function
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20. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.5
Continuity for a Composite Function
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21. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.2
If f is continuous at every point on an open interval
(a, b), we say that f is continuous on (a, b). Following the figure, we say that f is continuous on the closed interval
[a, b], if f is continuous on the open interval
(a, b) and
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22. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.2
Finally, if f is continuous on all of (−∞,∞), we simply say that f is continuous. (That is, when we don’t specify an interval, we mean continuous everywhere.)
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23. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.6
Continuity on a Closed Interval
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24. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.6
Continuity on a Closed Interval
Observe that f is defined only for
−2 ≤ x ≤ 2.
Note that f is the composition of two continuous functions and hence, is continuous for all x for which 4 − x2 > 0.
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25. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.6
Continuity on a Closed Interval
Since 4 − x2 > 0 for −2 < x < 2, we have that f is continuous for all x in the interval (−2, 2), by Theorem 4.1 and Corollary 4.1.
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26. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.6
Continuity on a Closed Interval
Finally, we test the endpoints to see that
and
so that f is continuous on the closed interval [−2, 2].
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27. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.4
(Intermediate Value Theorem)
Suppose that f is continuous on the closed interval [a, b] and W is any number between f (a) and f (b).
Then, there is a number c ∈ [a, b] for which f (c) = W.
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28. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
Intermediate Value Theorem 4.4
Theorem 4.4 says that if f is continuous on [a, b], then f must take on every value between f (a) and f (b) at least once.
That is, a continuous function cannot skip over any numbers between its values at the two endpoints. To do so, the graph would need to leap across the horizontal line y = W, something that continuous functions cannot do.
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29. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.2
Suppose that f is continuous on [a, b] and f (a) and f (b) have opposite signs [i.e., f (a) · f (b) < 0]. Then, there is at least one number
c ∈ (a, b) for which f (c) = 0.
(Recall that c is then a zero of f .)
Notice that Corollary 4.2 is simply the special case of the Intermediate Value Theorem
where W = 0.
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30. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.8
Finding Zeros by the Method of Bisections
(Since f is a polynomial of degree 5, we don’t have any formulas for finding its zeros. The only alternative then, is to approximate the zeros.)
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31. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.8
Finding Zeros by the Method of Bisections
Corollary 4.2 suggests a simple yet effective method, called the method of bisections.
Taking the midpoint of the interval [0, 1], since f (0.5) ≈ −0.469 < 0 and
f (0) = 3 > 0, there must be a zero between 0 and 0.5.
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32. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.8
Finding Zeros by the Method of Bisections
Next, the midpoint of [0, 0.5] is
0.25 and f (0.25) ≈ > 0, so that the zero is in the interval (0.25, 0.5).
We continue in this way to narrow down the interval in which there’s a zero
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33. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.8
Finding Zeros by the Method of Bisections
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34. CONTINUITY AND ITS CONSEQUENCES
1.4
CONTINUITY AND ITS CONSEQUENCES
4.8
Finding Zeros by the Method of Bisections
Continuing this process through 20 more steps leads to the approximate zero x = , which is accurate to at least eight decimal places.
The other zeros can be found in a similar fashion.
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