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Chapter 2: Problem 1
Suppose \(x\) lies in the interval (1,3) with \(x \neq 2 .\) Find the smallest positive value of \(\delta\) such that the inequality \(0<|x-2|<\delta\) is true.
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a. Find functions \(f\) and \(g\) such that each function is continuous at \(0\), but \(f \circ g\) is not continuous at \(0 .\) b. Explain why examples satisfying part (a) do not contradict Theorem 11.
Evaluate \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x)\). Then state the horizontal asymptote(s) of \(f\). Confirm your findings by plotting \(f\). $$f(x)=\frac{2 e^{x}+3 e^{2 x}}{e^{2 x}+e^{3 x}}$$
Assume the functions \(f, g,\) and \(h\) satisfy the inequality \(f(x) \leq g(x) \leq h(x)\) for all values of \(x\) near \(a\) except possibly at \(a\). Prove that if \(\lim _{x \rightarrow a} f(x)=\lim _{x \rightarrow a} h(x)=L\), then \(\lim _{x \rightarrow a} g(x)=L\).
Evaluate the following limits. $$\lim _{x \rightarrow 0} \frac{\cos x-1}{\sin ^{2} x}$$
A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence \(\\{2,4,6,8, \ldots\\}\) is specified by the function \(f(n)=2 n\), where \(n=1,2,3, \ldots .\) The limit of such a sequence is \(\lim _{n \rightarrow \infty} f(n)\), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences, or state that the limit does not exist. \(\left\\{0, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots .\right\\},\) which is defined by \(f(n)=\frac{n-1}{n},\) for \(n=1,2,3, \ldots\)
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