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Chapter 4: Problem 8

Let \(f\) be defined on \((0, \infty)\) to \(\mathbb{R}\). Prove that \(\lim _{x \rightarrow \infty} f(x)=L\) if and only if \(\lim _{x \rightarrow 0+} f(1 / x)=L\).

Short Answer

Expert verified
The statement that \(\lim _{x \rightarrow \infty} f(x)=L\) if and only if \(\lim _{x \rightarrow 0+} f(1 / x)=L\) stands proved from the previous steps.

Step by step solution

01

Only If Scenario

Assume \(\lim _{x \rightarrow \infty} f(x)=L\). For any given \(\epsilon>0\), there exists some \(M>0\) such that if \(x>M\), \(|f(x)-L|;\) will be less than \(\epsilon\) (as the definition of a limit). Now consider \(y=1/x\). If \(x>M\), \(y<1/M\). So if \(y<1/M\) (which means \(x>1/y\)), then \(|f(1/y)-L|\) will also be less than \(\epsilon\). This proves \( \lim _{x \rightarrow 0+} f(1 / x)=L\).
02

If Scenario

Assume \(\lim _{x \rightarrow 0+} f(1 / x)=L\). For any given \(\epsilon>0\), there exists \(0<\delta<1\) where \(\delta>0\) such that if \(01/\delta\). So if \(y>1/\delta\) (which means \(x<1/y\)), \(|f(y)-L|\) will be less than \(\epsilon\). This proves \(\lim _{x \rightarrow \infty} f(x)=L\).
03

Finishing the proof

As we have shown that \(\lim _{x \rightarrow \infty} f(x)=L\) if and only if \(\lim _{x \rightarrow 0+} f(1 / x)=L\), we can conclude that our original statement proves to be true.

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Most popular questions from this chapter

Let \(c \in \mathbb{R}\) and let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be such that \(\lim _{x \rightarrow c}(f(x))^{2}=L\). (a) Show that if \(L=0\), then \(\lim f(x)=0\) (b) Show by example that if \(L^{x \rightarrow c} \neq 0\), then \(f\) may not have a limit at \(c\).

Let \(I\) be an interval in \(\mathbb{R}\). let \(f: I \rightarrow \mathbb{R}\), and let \(c \in I\). Suppose there exist constants \(K\) and \(L\) such that \(|f(x)-L| \leq K|x-c|\) for \(x \in I .\) Show that \(\lim _{x \rightarrow c} f(x)=L\)

Use the definition of limit to show that (a) \(\lim _{x \rightarrow 2}\left(x^{2}+4 x\right)=12\), (b) \(\lim _{x \rightarrow-1} \frac{x+5}{2 x+3}=4\).

Let \(f, g\) be defined on \(A \subseteq \mathbb{R}\) to \(\mathbb{R}\), and let \(c\) be a cluster point of \(A .\) Suppose that \(f\) is bounded on a neighborhood of \(c\) and that \(\lim _{x \rightarrow c} g=0 .\) Prove that \(\lim _{x \rightarrow c} f g=0\)

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 \(\lim _{x \rightarrow c}|f|=\left|\lim _{x \rightarrow c} f\right|\)
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