Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 13 - Section 13.4 - Limits at Infinity; Limits of Sequences - 13.4 Exercises - Page 930: 38

Answer

See explanations.

Work Step by Step

(a) Assume $$\lim_{x\to\infty}f(x)=L$$ Let $t=\frac{1}{x}$, we have: $$\lim_{x\to\infty}t=0^+$$ Replace the variable $x$ with $t$ in the first limit, we have: $$\lim_{t\to 0^+}f(\frac{1}{t})=L$$ Thus we have: $$\lim_{x\to\infty}f(x)=\lim_{t\to 0^+}f(\frac{1}{t})$$ (b) Similarly, assume $$\lim_{x\to -\infty}f(x)=K$$ Let $t=\frac{1}{x}$, we have: $$\lim_{x\to -\infty}t=0^-$$ Replace the variable $x$ with $t$ in the starting limit, we have: $$\lim_{t\to 0^-}f(\frac{1}{t})=K$$ Thus we have: $$\lim_{x\to -\infty}f(x)=\lim_{t\to 0^-}f(\frac{1}{t})$$
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