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Chapter 1: Problem 29
Find the limit (if it exists). $$ \lim _{x \rightarrow 5} \frac{x-5}{x^{2}-25} $$
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(a) Prove that if \(\lim _{x \rightarrow c}|f(x)|=0,\) then \(\lim _{x \rightarrow c} f(x)=0\). (Note: This is the converse of Exercise \(74 .)\) (b) Prove that if \(\lim _{x \rightarrow c} f(x)=L,\) then \(\lim _{x \rightarrow c}|f(x)|=|L|\). [Hint: Use the inequality \(\|f(x)|-| L\| \leq|f(x)-L| .]\)
Use a graphing utility to graph the given function and the equations \(y=|x|\) and \(y=-|x|\) in the same viewing window. Using the graphs to visually observe the Squeeze Theorem, find \(\lim _{x \rightarrow 0} f(x)\). $$ h(x)=x \cos \frac{1}{x} $$
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 \(x \rightarrow 25^{-}\).
Explain why the function has a zero in the given interval. $$ \begin{array}{lll} \text { Function } & \text { Interval } \\ f(x)=x^{3}+3 x-2 & {[0,1]} \\ \end{array} $$
(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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