Find each limit in Problems 37–60. Note that L’Hôpital’s rule does not apply to every problem , and some problems will require more than one application of L’Hôpital’s rule. 49. lim x → 2 x 3 − 12 x + 16 x 3 − 6 x 2 + 12 x − 8
Find each limit in Problems 37–60. Note that L’Hôpital’s rule does not apply to every problem , and some problems will require more than one application of L’Hôpital’s rule. 49. lim x → 2 x 3 − 12 x + 16 x 3 − 6 x 2 + 12 x − 8
Solution Summary: The author evaluates the value of the limit undersetxto.
Find each limit in Problems 37–60. Note that L’Hôpital’s rule does not apply to every problem, and some problems will require more than one application of L’Hôpital’s rule.
49.
lim
x
→
2
x
3
−
12
x
+
16
x
3
−
6
x
2
+
12
x
−
8
ball is drawn from one of three urns depending on the outcomeof a roll of a dice. If the dice shows a 1, a ball is drawn from Urn I, whichcontains 2 black balls and 3 white balls. If the dice shows a 2 or 3, a ballis drawn from Urn II, which contains 1 black ball and 3 white balls. Ifthe dice shows a 4, 5, or 6, a ball is drawn from Urn III, which contains1 black ball and 2 white balls. (i) What is the probability to draw a black ball? [7 Marks]Hint. Use the partition rule.(ii) Assume that a black ball is drawn. What is the probabilitythat it came from Urn I? [4 Marks]Total marks 11 Hint. Use Bayes’ rule
Let X be a random variable taking values in (0,∞) with proba-bility density functionfX(u) = 5e^−5u, u > 0.Let Y = X2 Total marks 8 . Find the probability density function of Y .
Let P be the standard normal distribution, i.e., P is the proba-bility measure on R, B(R) given bydP(x) = 1√2πe− x2/2dx.Consider the random variablesfn(x) = (1 + x2) 1/ne^(x^2/n+2) x ∈ R, n ∈ N.Using the dominated convergence theorem, prove that the limitlimn→∞E(fn)exists and find it
Chapter 4 Solutions
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
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