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Chapter 4: Problem 8
Show that \(\lim _{x \rightarrow c} \sqrt{x}=\sqrt{c}\) for any \(c>0\).
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Let \(A \subseteq \mathbb{R}\), let \(f: A \rightarrow \mathbb{R}\) and let \(c \in \mathbb{R}\) be a cluster point of \(A .\) If \(\lim _{\tau \rightarrow c} f\) exists, and if \(|f|\) denotes the function defined for \(x \in A\) by \(|f|(x):=|f(x)|\), prove that \(\lim _{x \rightarrow c}|f|=\left|\lim _{x \rightarrow c} f\right|\)
Let \(f\) be defined on \((0, \infty)\) to \(\mathbb{R}\). Prove that \(\lim _{x \rightarrow \infty} f(x)=L\) if and only if \(\lim _{x \rightarrow 0+} f(1 / x)=L\).
Suppose that \(f\) and \(g\) have limits in \(\mathbb{R}\) as \(x \rightarrow \infty\) and that \(f(x) \leq g(x)\) for all \(x \in(a, \infty)\). Prove that \(\lim _{x \rightarrow \infty} f \leq \lim _{x \rightarrow \infty} g\).
Let \(f, g\) be defined on \(A\) to \(\mathbb{R}\) and let \(c\) be a cluster point of \(A\). (a) Show that if both \(\lim _{x \rightarrow c} f\) and \(\lim _{x \rightarrow c}(f+g)\) cxist, then \(\lim _{x \rightarrow c} g\) exists. (b) If \(\lim _{x \rightarrow c} f\) and \(\lim _{x \rightarrow c} f g\) exist, does it follow that \(\lim _{x \rightarrow c} g\) exists?
Let \(c\) be a cluster point of \(A \subseteq \mathbb{R}\) and let \(f: A \rightarrow \mathbb{R}\). Prove that \(\lim _{x \rightarrow c} f(x)=L\) if and only if \(\lim _{x \rightarrow c}|f(x)-L|=0\)
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