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Chapter 2: Problem 17
Determine the following limits. $$\lim _{x \rightarrow \infty} x^{-6}$$
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Use the following definition for the nonexistence of a limit. Assume \(f\) is defined for all values of \(x\) near a, except possibly at a. We say that \(\lim _{x \rightarrow a} f(x) \neq L\) if for some \(\varepsilon>0\) there is no value of \(\delta>0\) satisfying the condition $$|f(x)-L|<\varepsilon \quad \text { whenever } \quad 0<|x-a|<\delta$$ Let $$f(x)=\left\\{\begin{array}{ll} 0 & \text { if } x \text { is rational } \\ 1 & \text { if } x \text { is irrational. } \end{array}\right.$$ Prove that \(\lim _{x \rightarrow a} f(x)\) does not exist for any value of \(a\). (Hint: Assume \(\lim _{x \rightarrow a} f(x)=L\) for some values of \(a\) and \(L\) and let \(\varepsilon=\frac{1}{2}\).)
Analyzing infinite limits graphically Graph the function \(y=\tan x\) with the window \([-\pi, \pi] \times[-10,10] .\) Use the graph to analyze the following limits. a. \(\lim _{x \rightarrow \pi / 2^{+}} \tan x\) b. \(\lim _{x \rightarrow \pi / 2^{-}} \tan x\) d. \(\quad \lim _{n}\) tan \(x\) c. \(\lim _{x \rightarrow-\pi / 2^{+}} \tan x\) d. \(\lim _{x \rightarrow-\pi / 2^{-}} \tan x\)
The limit at infinity \(\lim _{x \rightarrow \infty} f(x)=L\) means that for any \(\varepsilon>0,\) there exists \(N>0\) such that $$|f(x)-L|<\varepsilon \quad \text { whenever } \quad x>N$$ Use this definition to prove the following statements. $$\lim _{x \rightarrow \infty} \frac{10}{x}=0$$
We say that \(\lim _{x \rightarrow \infty} f(x)=\infty\) if for any positive number \(M,\) there is \(a\) corresponding \(N>0\) such that $$f(x)>M \quad \text { whenever } \quad x>N$$ Use this definition to prove the following statements. $$\lim _{x \rightarrow \infty} \frac{x}{100}=\infty$$
Use the following definition for the nonexistence of a limit. Assume \(f\) is defined for all values of \(x\) near a, except possibly at a. We say that \(\lim _{x \rightarrow a} f(x) \neq L\) if for some \(\varepsilon>0\) there is no value of \(\delta>0\) satisfying the condition $$|f(x)-L|<\varepsilon \quad \text { whenever } \quad 0<|x-a|<\delta$$ Prove that \(\lim _{x \rightarrow 0} \frac{|x|}{x}\) does not exist.
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