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Chapter 4: Problem 8
Show that \(\lim _{x \rightarrow c} x^{3}=c^{3}\) for any \(c \in \mathbb{R}\).
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Suppose that \(\lim _{x \rightarrow c} f(x)=L\) where \(L>0\), and that \(\lim _{x \rightarrow c} g(x)=\infty\). Show that \(\lim _{x \rightarrow c} f(x) g(x)=\infty\). If \(L=0\), show by example that this conclusion may fail.
Let \(f:=\mathbb{R} \rightarrow \mathbb{R}\) and let \(c \in \mathbb{R}\). Show that \(\lim _{x \rightarrow c} f(x)=L\) if and only if \(\lim _{x \rightarrow 0} f(x+c)=L\).
Show that if \(f:(a, \infty) \rightarrow \mathbb{R}\) is such that \(\lim _{x \rightarrow \infty} x f(x)=L\) where \(L \in \mathbb{R}\), then \(\lim _{x \rightarrow \infty} f(x)=0\)
Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by setting \(f(x):=x\) if \(x\) is rational, and \(f(x)=0\) if \(x\) is irrational. (a) Show that \(f\) has a limit at \(x=0\). (b) Use a sequential argument to show that if \(c \neq 0\), then \(f\) does not have a limit at \(c\).
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 \(\lim _{x \rightarrow \infty} f \leq \lim _{x \rightarrow \infty} g\).
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