Calculus (3rd Edition)

Published by W. H. Freeman

Chapter 2 - Limits - 2.6 Trigonometric Limits - Exercises - Page 77: 5

Answer

(a) The given inequality does not provide sufficient information to determine $\lim _{x \rightarrow 1} f(x)$ (b) $\lim _{x \rightarrow 1} f(x)=1$ (c) $\lim _{x \rightarrow 1} f(x)=3$

Work Step by Step

(a) Because $\lim _{x \rightarrow 1}(4 x-5)=-1 \neq 1=\lim _{x \rightarrow 1} x^{2},$ the given inequality does not provide sufficient information to determine $\lim _{x \rightarrow 1} f(x)$ (b) Because $\lim _{x \rightarrow 1}(2 x-1)=1=\lim _{x \rightarrow 1} x^{2},$ it follows from the Squeeze Theorem that $\lim _{x \rightarrow 1} f(x)=1$ (c) Because $\lim _{x \rightarrow 1}\left(4 x-x^{2}\right)=3=\lim _{x \rightarrow 1}\left(x^{2}+2\right),$ it follows from the Squeeze Theorem that $\lim _{x \rightarrow 1} f(x)=3$

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