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Chapter 1: Problem 74
Prove that the product of an odd function and an even function is odd.
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Prove that if \(\lim _{x \rightarrow c} f(x)\) exists and \(\lim _{x \rightarrow c}[f(x)+g(x)]\) does not exist, then \(\lim _{x \rightarrow c} g(x)\) does not exist.
Describe the difference between a discontinuity that is removable and one that is nonremovable. In your explanation, give examples of the following. (a) A function with a nonremovable discontinuity at \(x=2\) (b) A function with a removable discontinuity at \(x=-2\) (c) A function that has both of the characteristics described in parts (a) and (b)
In Exercises 129 and \(130,\) verify each identity (a) \(\operatorname{arccsc} x=\arcsin \frac{1}{x}, \quad|x| \geq 1\) (b) \(\arctan x+\arctan \frac{1}{x}=\frac{\pi}{2}, \quad x>0\)
True or False? In Exercises \(50-53\), determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) has a vertical asymptote at \(x=0,\) then \(f\) is undefined at \(x=0\)
Sketch the graph of any function \(f\) such that \(\lim _{x \rightarrow 3^{+}} f(x)=1\) and \(\quad \lim _{x \rightarrow 3^{-}} f(x)=0\). Is the function continuous at \(x=3\) ? Explain.
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