91Ó°ÊÓ

Sketch the interval \((a, b)\) on the \(x\) -axis with the point \(c\) inside. Then find a value of \(\delta>0\) such that for all \(x, 0<|x-c|<\delta \Rightarrow a

If \(\lim _{x \rightarrow 1} f(x)=5,\) must \(f\) be defined at \(x=1 ?\) If it is, must \(f(1)=5 ?\) Can we conclude anything about the values of \(f\) at \(x=1 ?\) Explain.

Find the limits. $$\lim _{\theta \rightarrow-\infty} \frac{\cos \theta}{3 \theta}$$

Find the limits. $$\lim _{x \rightarrow 1^{+}} \sqrt{\frac{x-1}{x+2}}$$

At what points are the functions in Exercises \(13-30\) continuous? $$f(x)=\left\\{\begin{array}{ll} \frac{x^{3}-8}{x^{2}-4}, & x \neq 2, x \neq-2 \\\ 3, & x=2 \\ 4, & x=-2 \end{array}\right.$$

Define \(g(3)\) in a way that extends \(g(x)=\left(x^{2}-9\right) /(x-3)\) to be continuous at \(x=3\)

Given \(\epsilon>0,\) find an interval \(I=(5,5+\delta), \delta>0,\) such that if \(x\) lies in \(I\), then \(\sqrt{x-5}<\epsilon .\) What limit is being verified and what is its value?

A wrong statement about limits Show by example that the following statement is wrong. The number \(L\) is the limit of \(f(x)\) as \(x\) approaches \(c\) if \(f(x)\) gets closer to \(L\) as \(x\) approaches \(c\) Explain why the function in your example does not have the given value of \(L\) as a limit as \(x \rightarrow c\)

Is it true that a continuous function that is never zero on an interval never changes sign on that interval? Give reasons for your answer.

A fixed point theorem Suppose that a function \(f\) is continuous on the closed interval [0,1] and that \(0 \leq f(x) \leq 1\) for every \(x\) in \([0,1] .\) Show that there must exist a number \(c\) in [0,1] such that \(f(c)=c(c \text { is called a fixed point of } f)\)