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Chapter 1: Problem 4
Find the limit. $$ \lim _{x \rightarrow-3}(3 x+2) $$
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$$ \begin{aligned} &\text { Prove that if } f \text { and } g \text { are one-to-one functions, then }\\\ &(f \circ g)^{-1}(x)=\left(g^{-1} \circ f^{-1}\right)(x). \end{aligned} $$
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) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{4}}\)
Does every rational function have a vertical asymptote? Explain.
Write the expression in algebraic form. \(\cos (\operatorname{arccot} x)\)
After an object falls for \(t\) seconds, the speed \(S\) (in feet per second) of the object is recorded in the table. $$ \begin{array}{|l|c|c|c|c|c|c|c|} \hline t & 0 & 5 & 10 & 15 & 20 & 25 & 30 \\ \hline S & 0 & 48.2 & 53.5 & 55.2 & 55.9 & 56.2 & 56.3 \\ \hline \end{array} $$ (a) Create a line graph of the data. (b) Does there appear to be a limiting speed of the object? If there is a limiting speed, identify a possible cause.
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