Chapter 2: Worksheet 3 ANSWERS MA 131
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u = x3, h(x) = sin u or
g(x) = x3, f (x) = sin x
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u = sin x, h = u3
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u = sin x, h = 3u
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u = x3 +3, h = sin u
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u = (x4 + x2 +1), h = u7
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u = x + 1, v = ln u, h = tan v
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u = x2, v = sin u, h = 2v
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u = sin x, v = sin u, h = sin v
- 0
- 0 (see Figure 1 and 2). Note
sinπ/2 = 1.0 and
sin(sin(sinπ/2)) = 0.745624.
Since π/2 produces the largest value, once we have
the result there, it isn't hard to realize that the function goes to 0 at every x as we
take more and more compositions.
Figure:
4 compositions of sin with itself
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Figure:
128 compositions of sin with itself
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- Merely evaluating the function to get the limit is justified when the function is continuous. If you don't see this important point, study the definition of continuity carefully and/or ask your instructor.
- Rational functions are continuous except at points where the denominator vanishes. Hence we may just ``plug in'' for this example:
=
=
- Here we have the favorite case in calculus where numerator and denominator both vanish at a. Because of the denominator, the function is not continuous at x = 5. However, in evaluating the limit, we do not have to consider this point, so we may employ factoring for x≠5 to get
(x + 5) = 10.
- No problem, just ``plug in'':

=
=
- 1 (For example, make a Table or use the Program of the Week)
- 0
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e
2.71828