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Chapter 2: Problem 21
Determine the following limits. $$\lim _{x \rightarrow-\infty}\left(-3 x^{16}+2\right)$$
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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 population of a colony of squirrels is given by \(p(t)=\frac{1500}{3+2 e^{-0.1 t}}\).
Electric field The magnitude of the electric field at a point \(x\) meters from the midpoint of a \(0.1-\mathrm{m}\) line of charge is given by \(E(x)=\frac{4.35}{x \sqrt{x^{2}+0.01}}\) (in units of newtons per coulomb, \(\mathrm{N} / \mathrm{C}\) ). Evaluate \(\lim _{x \rightarrow 10} E(x)\).
Suppose \(f\) is continuous at \(a\) and assume \(f(a)>0 .\) Show that there is a positive number \(\delta>0\) for which \(f(x)>0\) for all values of \(x\) in \((a-\delta, a+\delta)\). (In other words, \(f\) is positive for all values of \(x\) sufficiently close to \(a .\) )
Analyzing infinite limits graphically Graph the function \(y=\sec x \tan x\) with the window \([-\pi, \pi] \times[-10,10] .\) Use the graph to analyze the following limits. a. \(\lim _{x \rightarrow \pi / 2^{+}} \sec x \tan x\) b. \(\lim _{x \rightarrow \pi / 2^{-}} \sec x \tan x\) c. \(\lim _{x \rightarrow-\pi / 2^{+}} \sec x \tan x\) d. \(\lim _{x \rightarrow-\pi / 2^{-}} \sec x \tan x\)
Evaluate the following limits. $$\lim _{x \rightarrow \pi} \frac{\cos ^{2} x+3 \cos x+2}{\cos x+1}$$
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