91Ó°ÊÓ

The limit of the greatest integer function as \(x\) approaches 0 from the left is \(-1 .\)

Volume Use the Intermediate Value Theorem to show that for all spheres with radii in the interval \([5,8],\) there is one with a volume of 1500 cubic centimeters.

Show that the Dirichlet function $$f(x)=\left\\{\begin{array}{ll}{0,} & {\text { if } x \text { is rational }} \\\ {1,} & {\text { if } x \text { is irrational }}\end{array}\right.$$ is not continuous at any real number.

Graphical Reasoning Consider \(f(x)=\frac{\sec x-1}{2}\) (a) Find the domain of \(f\) (b) Use a graphing utility to graph \(f .\) Is the domain of \(f\)obvious from the graph? If not, explain. (c) Use the graph of \(f\) to approximate $$\lim _{x \rightarrow 0} f(x)$$ (d) Confirm your answer to part (c) analytically.

Making a Function Continuous Find all values of \(c\) such that \(f\) is continuous on \((-\infty, \infty)\) \(f(x)=\left\\{\begin{array}{ll}{1-x^{2},} & {x \leq c} \\ {x,} & {x>c}\end{array}\right.\)