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Chapter 1: Problem 6
Find the limit. $$ \lim _{x \rightarrow-3} \frac{2}{x+2} $$
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In Exercises \(35-38\), use a graphing utility to graph the function and determine the one-sided limit. $$ \begin{array}{l} f(x)=\frac{x^{2}+x+1}{x^{3}-1} \\ \lim _{x \rightarrow 1^{+}} f(x) \end{array} $$
Determine conditions on the constants \(a, b,\) and \(c\) such that the graph of \(f(x)=\frac{a x+b}{c x-a}\) is symmetric about the line \(y=x\).
Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=x^{3}-x^{2}+x-2, \quad[0,3], \quad f(c)=4 $$
Rate of Change A 25 -foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a rate \(r\) of \(r=\frac{2 x}{\sqrt{625-x^{2}}} \mathrm{ft} / \mathrm{sec}\) where \(x\) is the distance between the ladder base and the house. (a) Find \(r\) when \(x\) is 7 feet. (b) Find \(r\) when \(x\) is 15 feet. (c) Find the limit of \(r\) as \(x \rightarrow 25^{-}\).
True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=g(x)\) for \(x \neq c\) and \(f(c) \neq g(c),\) then either \(f\) or \(g\) is not continuous at \(c\).
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