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(d) Using the formal - 6 definition of the limit, prove that x lim →∞ In x Hint: you may need to use the result of part (c) and the logarithm is an increasing function.

(d) Using the formal - 6 definition of the limit, prove that x lim →∞ In x Hint: you may need to use the result of part (c) and the logarithm is an increasing function.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter4: Calculating The Derivative
Section4.5: Derivatives Of Logarithmic Functions
Problem 56E: This exercise shows another way to derive the formula for the derivative of the natural logarithm...
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(d) Using the formal e - definition of the limit, prove that lim x→∞ ln x Hint: you may need to use the result of part (c) and the logarithm is an increasing function.
Transcribed Image Text:(d) Using the formal e - definition of the limit, prove that lim x→∞ ln x Hint: you may need to use the result of part (c) and the logarithm is an increasing function.
2. Here you will prove that lim = ∞ without any use of L'Hopital's Rule. →∞ In r (a) For all positive integers n ≥ 1, prove that 2" > 2n. (b) For all real numbers x ≥ 1, prove that 2r. Hint:Define n = [r]. (c) Recall e = 2.71828... is Euler's constant and e² is the exponential function. Since e > 2, you may assume that lim ()* = ∞. Show that et lim == ∞0. xx x Hint: you may use the result of part (b) and a new Squeeze Theorem in one of the lectures.
Transcribed Image Text:2. Here you will prove that lim = ∞ without any use of L'Hopital's Rule. →∞ In r (a) For all positive integers n ≥ 1, prove that 2" > 2n. (b) For all real numbers x ≥ 1, prove that 2r. Hint:Define n = [r]. (c) Recall e = 2.71828... is Euler's constant and e² is the exponential function. Since e > 2, you may assume that lim ()* = ∞. Show that et lim == ∞0. xx x Hint: you may use the result of part (b) and a new Squeeze Theorem in one of the lectures.
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