Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - 11.1 Sequences - Exercises - Page 538: 62



Work Step by Step

When $\lim\limits_{x \to \infty} f(x)$ exists, then the sequence $a_n=f(n)$ converges to the same limit, so we have: $ \lim\limits_{n \to \infty} a_n=\lim\limits_{x \to \infty} f(x)$ Now, $ \lim\limits_{n \to \infty} a_n=\lim\limits_{x \to \infty} (1+\dfrac{1}{x^2})^x \\=\lim\limits_{x \to \infty} (1+\dfrac{1}{x^2})^{x^2 \times \dfrac{1}{x}}$ Since, $\lim\limits_{x \to \infty} (1+\dfrac{1}{x^2})^{x^2}=e$ Thus, $ \lim\limits_{n \to \infty} a_n =e^{\lim\limits_{x \to \infty}\frac{1}{x}} \\ =e^0 \\=1$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.