Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 7 - Exponential Functions - 7.5 Compound Interest and Present Value - Exercises - Page 356: 24


$ e^3 $

Work Step by Step

Let $ x=\frac{n}{3}$, then we have $$\lim _{n \rightarrow \infty}\left(1+\frac{3}{n}\right)^{n}=\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{3x}\\ =\left(\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}\right)^3=e^3,$$ where we used the formula $\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=e $.
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