Solved: Exercises 54 and 55 use LHpitals rule from calculus
Chapter 11, Problem 54(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 and mathematical induction to prove that for all integers \(n \geq 1\),
\(\lim _{x \rightarrow \infty} \frac{x^{n}}{b^{x}}=0\) .
b. Use the result of part (a) and the definitions of limit and of O-notation to prove that \(x^{n}\) is \(O\left(b^{x}\right)\) for any integer \(n \geq 1\).
Text Transcription:
n geq 1
lim _x rightarrow infty frac x^n b^x=0
x^n
O(b^x)
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