Solved: Exercises 54 and 55 use LHpitals rule from calculus
Chapter 11, Problem 55(choose chapter or problem)
Exercises 54 and 55 use L'Hôpital's rule from calculus.
a. Let b be any real number greater than 1 . Use L'Hôpital's rule to prove that for all integers \(n \geq 1\),
\(\lim _{x \rightarrow \infty} \frac{\log _{b} x}{x^{1 / n}}=0\) .
b. Use the result of part (a) and the definitions of limit and of O-notation to prove that \(\log _{b} x\) is \(O\left(x^{1 / n}\right)\) for any integer \(n \geq 1\).
Text Transcription:
n geq 1
lim _x rightarrow infty frac log _b x x^1 / n=0
log _b x
O(x^1 / n)
n geq 1
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