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Chapter 1: Problem 35
Find the limit (if it exists). $$ \lim _{\Delta x \rightarrow 0} \frac{2(x+\Delta x)-2 x}{\Delta x} $$
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In Exercises \(25-34,\) find the limit. $$ \lim _{x \rightarrow 0^{+}} \frac{2}{\sin x} $$
Describe how the functions \(f(x)=3+\llbracket x \rrbracket\) and \(g(x)=3-\llbracket-x \rrbracket\) differ.
Use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval [0, 1]. Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to approximate the zero accurate to four decimal places. $$ f(x)=x^{3}+3 x-3 $$
Rate of Change A patrol car is parked 50 feet from a long warehouse (see figure). The revolving light on top of the car turns at a rate of \(\frac{1}{2}\) revolution per second. The rate \(r\) at which the light beam moves along the wall is \(r=50 \pi \sec ^{2} \theta \mathrm{ft} / \mathrm{sec}\) (a) Find \(r\) when \(\theta\) is \(\pi / 6\). (b) Find \(r\) when \(\theta\) is \(\pi / 3\). (c) Find the limit of \(r\) as \(\theta \rightarrow(\pi / 2)^{-}\)
$$ \begin{aligned} &\text { Prove that if } f \text { and } g \text { are one-to-one functions, then }\\\ &(f \circ g)^{-1}(x)=\left(g^{-1} \circ f^{-1}\right)(x). \end{aligned} $$
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