Chapter 1: Problem 11
Find the limit. $$ \lim _{x \rightarrow 0} \sec 2 x $$
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Numerical and Graphical Analysis Use a graphing utility to complete the table for each function and graph each function to estimate the limit. What is the value of the limit when the power on \(x\) in the denominator is greater than \(3 ?\) $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 1 & 0.5 & 0.2 & 0.1 & 0.01 & 0.001 & 0.0001 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & & \\ \hline \end{array} $$ (a) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x}\) (b) \(\lim _{x \rightarrow 0^{-}} \frac{x-\sin x}{x^{2}}\) (c) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{3}}\) (d) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{4}}\)
In the context of finding limits, discuss what is meant by two functions that agree at all but one point.
Prove that if \(f\) is continuous and has no zeros on \([a, b],\) then either \(f(x)>0\) for all \(x\) in \([a, b]\) or \(f(x)<0\) for all \(x\) in \([a, b]\)
In Exercises \(25-34,\) find the limit. $$ \lim _{x \rightarrow 0^{+}} \frac{2}{\sin x} $$
(a) Let \(f_{1}(x)\) and \(f_{2}(x)\) be continuous on the closed interval \([a,
b]\). If \(f_{1}(a)
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