Chapter 1: Problem 23
Find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? $$ f(x)=x^{2}-2 x+1 $$
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Writing Use a graphing utility to graph \(f(x)=x, \quad g(x)=\sin x, \quad\) and \(\quad h(x)=\frac{\sin x}{x}\) in the same viewing window. Compare the magnitudes of \(f(x)\) and \(g(x)\) when \(x\) is "close to" \(0 .\) Use the comparison to write \(a\) short paragraph explaining why \(\lim _{x \rightarrow 0} h(x)=1\).
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \arcsin ^{2} x+\arccos ^{2} x=1 $$
In Exercises \(131-134,\) sketch the graph of the function. Use a graphing utility to verify your graph. $$ f(x)=\arcsin (x-1) $$
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 \(c\).
Use the Intermediate Value Theorem to show that for all spheres with radii in the interval [1,5] , there is one with a volume of 275 cubic centimeters.
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