
(a)
The mathematical proof of the statement stating that using L'Hopital Rule, , though not defined at x = 0, can be made continuous by assigning the value

Answer to Problem 53E
Solution: The mathematical proof of the statement stating that using L'Hopital Rule, , though not defined at x = 0, can be made continuous by assigning the value is derived.
Explanation of Solution
Explanation:
Given: for a>0
Calculation:
Step 1: The function , is continuous if .
Step 2: We verify this limit using L'Hopital Rule:
Therefore, f is continuous.
Conclusion: The statement stating that using L'Hopital Rule, , though not defined at x = 0, can be made continuous by assigning the value is mathematically proved.
(b)
The mathematical proof of the statement stating that for using triangle inequality and the mathematical proof of the statement stating that I(a) converges by applying the Comparison Theorem

Answer to Problem 53E
Solution: Both the mathematical proof of the statement stating that for using triangle inequality and the mathematical proof of the statement stating that I(a) converges by applying the Comparison Theorem are derived.
Explanation of Solution
Given: , , , a > 0
Calculation:
Step 1: We now show that the following integral converges:
(a>0)
Since, then for x > 0
If x > 1 we have,
That is for x > 1
Step 2: Also, since we have for x > 1
Thus, we get
Step 3: Hence, from Step 1 and Step 2, we get
Step 4: We now show that the integral of the right hand side converges:
Since the integral converges, we conclude from Step 3 and the Comparison Test for Improper Integral that
also converges for a > 0.
Conclusion: Both the statement stating that for using triangle inequality and the statement stating that I(a) converges by applying the Comparison Theorem are mathematically proved.
(c)
The mathematical proof of the equation

Answer to Problem 53E
Solution: The mathematical proof of the equation is derived
Explanation of Solution
Given: , a>0
Calculation:
Step 1: We compute the inner integral with respect to y:
Step 2: Hence,
Conclusion: The equation is mathematically proved.
(d)
By interchanging the order of

Answer to Problem 53E
Solution: By interchanging the order of integration, the mathematical proof of the equation
is derived.
Explanation of Solution
Given: , a>0
Calculation:
Step 1: By the definition of the improper integral,
Step 2: We compute the double integral. Using Fubini's Theorem we may compute the iterated integral using the reversed order of integration. That is,
Combining with Step 1, we get,
Conclusion: By interchanging the order of integration, the equation is mathematically proved.
(e)
The mathematical proof of the statement stating the limit in is zero by using the Comparison Theorem.

Answer to Problem 53E
Solution: The mathematical proof of the statement stating the limit in is zero by using the Comparison Theorem is derived.
Explanation of Solution
Given: , a>0
Calculation:
Step 1: We consider the following possible cases:
Case 1: then in the interval of integration . As , we may assume that T > 0
Thus,
Hence,
By the limit and the Squeeze Theorem, we conclude that,
Case 2:
Then,
and in the interval of integration , therefore
(the function is decreasing).
Hence,
By the limit and the Squeeze Theorem, we conclude that
We thus showed that for all a > 0,
Conclusion: The statement stating the limit in is zero by using the Comparison Theorem is mathematically proved.
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Chapter 15 Solutions
Calculus: Early Transcendentals
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