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Chapter 2: Problem 16
Determine the following limits. $$\lim _{x \rightarrow-\infty} 3 x^{11}$$
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Suppose \(f(x)=\frac{p(x)}{q(x)}\) is a rational function, where \(p(x)=a_{m}
x^{m}+a_{m-1} x^{m-1}+\cdots+a_{2} x^{2}+a_{1} x+a_{0}\), \(q(x)=b_{n}
x^{n}+b_{n-1} x^{n-1}+\cdots+b_{2} x^{2}+b_{1} x+b_{0}, a_{m} \neq 0\), and
\(b_{n} \neq 0\).
a. Prove that if \(m=n,\) then \(\lim _{x \rightarrow \pm \infty}
f(x)=\frac{a_{m}}{b_{n}}\).
b. Prove that if \(m
Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$p(x)=\sec \left(\frac{\pi x}{2}\right), \text { for }|x|<2$$
Assume you invest \(\$ 250\) at the end of each year for 10 years at an annual interest rate of \(r .\) The amount of money in your account after 10 years is \(A=\frac{250\left((1+r)^{10}-1\right)}{r}\) Assume your goal is to have \(\$ 3500\) in your account after 10 years. a. Use the Intermediate Value Theorem to show that there is an interest rate \(r\) in the interval \((0.01,0.10)-\) between \(1 \%\) and \(10 \%-\) that allows you to reach your financial goal. b. Use a calculator to estimate the interest rate required to reach your financial goal.
If a function \(f\) represents a system that varies in time, the existence of \(\lim _{t \rightarrow \infty} f(t)\) means that the system reaches a steady state (or equilibrium). For the following systems, determine if a steady state exists and give the steady-state value. The amplitude of an oscillator is given by \(a(t)=2\left(\frac{t+\sin t}{t}\right)\).
Sketch a possible graph of a function \(f\) that satisfies all of the given conditions. Be sure to identify all vertical and horizontal asymptotes. $$\lim _{x \rightarrow 0^{+}} f(x)=\infty, \lim _{x \rightarrow 0^{-}} f(x)=-\infty, \lim _{x \rightarrow \infty} f(x)=1$$, $$\lim _{x \rightarrow-\infty} f(x)=-2$$
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