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Chapter 4: Problem 7

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 \(\lim _{x \rightarrow \infty} f \leq \lim _{x \rightarrow \infty} g\).

Short Answer

Expert verified
The limit of \(f\) as \(x \rightarrow \infty\) is less than or equal to the limit of \(g\) as \(x \rightarrow \infty\).

Step by step solution

01

Assume the contrary

Let's assume the contrary. Suppose that the limit of \(f\) is greater than the limit of \(g\) as \(x \rightarrow \infty\). This implies that there is a positive number \(c > 0\) such that \(\lim_{x \rightarrow \infty} f > \lim_{x \rightarrow \infty} g + c\). This means that for \(x\) sufficiently large, we have \(f(x) > g(x) + c\).
02

Show contradiction

Step 1's conclusion \(f(x) > g(x) + c\) contradicts the assumption that \(f(x) \leq g(x)\) for all \(x \in(a, \infty)\). Hence, it must be the case that our assumption was false, and so it must instead be true that \(\lim_{x \rightarrow \infty} f \leq \lim_{x \rightarrow \infty} g\).

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Most popular questions from this chapter

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.

Determine whether the following limits exist in \(\mathbb{K}\). (a) \(\lim _{x \rightarrow 0} \sin \left(1 / x^{2}\right) \quad(x \neq 0)\) (b) \(\lim _{x \rightarrow 0} x \sin \left(1 / x^{2}\right) \quad(x \neq 0)\) (c) \(\lim _{x \rightarrow 0} \operatorname{sgn} \sin (1 / x) \quad(x \neq 0)\), (d) \(\lim _{x \rightarrow 0} \sqrt{x} \sin \left(1 / x^{2}\right) \quad(x>0)\).

Evaluate the following limits, or show that they do not exist. (a) \(\lim _{x \rightarrow 1+} \frac{x}{x-1} \quad(x \neq 1)\), (b) \(\lim _{x \rightarrow 1} \frac{x}{x-1} \quad(x \neq 1)\), (c) \(\lim _{x \rightarrow 0^{+}}(x+2) / \sqrt{x} \quad(x>0)\), (d) \(\lim _{x \rightarrow \infty}(x+2) / \sqrt{x} \quad(x>0)\), (e) \(\lim _{x \rightarrow 0}(\sqrt{x+1}) / x \quad(x>-1)\), (f) \(\lim _{x \rightarrow \infty}(\sqrt{x+1}) / x \quad(x>0)\), (g) \(\lim _{x \rightarrow \infty} \frac{\sqrt{x}-5}{\sqrt{x}+3} \quad(x>0)\), (h) \(\lim _{x \rightarrow \infty} \frac{\sqrt{x}-x}{\sqrt{x}+x} \quad(x>0)\).

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 \(I:=(0, a)\) where \(a>0\), and let \(g(x):=x^{2}\) for \(x \in I\). For any points \(x, c \in I\), show that \(\left|g(x)-c^{2}\right| \leq 2 a|x-c| .\) Use this inequality to prove that \(\lim _{x \rightarrow c} x^{2}=c^{2}\) for any \(c \in I\).
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