Chapter 1: Problem 32
Sketch the graph of the function and state its domain. $$ f(x)=-2 \ln x $$
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Show that the function \(f(x)=\left\\{\begin{array}{ll}0, & \text { if } x \text { is rational } \\ k x, & \text { if } x \text { is irrational }\end{array}\right.\) is continuous only at \(x=0\). (Assume that \(k\) is any nonzero real number.)
Explain why the function has a zero in the given interval. $$ \begin{array}{lll} \text { Function } & \text { Interval } \\ g(t)=\left(t^{3}+2 t-2\right) \ln \left(t^{2}+4\right) & {[0,1]} \end{array} $$
In Exercises \(35-38\), use a graphing utility to graph the function and determine the one-sided limit. $$ \begin{array}{l} f(x)=\frac{x^{3}-1}{x^{2}+x+1} \\ \lim _{x \rightarrow 1^{-}} f(x) \end{array} $$
Prove that if \(f\) has an inverse function, then \(\left(f^{-1}\right)^{-1}=f\).
In the context of finding limits, discuss what is meant by two functions that agree at all but one point.
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