Chapter 13

Find the rule of the derivative of the function \(f .\) [The difference quotients of these functions were found and simplified in Exercises \(57-60 \text { of Section } 5.1 .]\) $$f(x)=\sqrt{x^{2}-x}$$

Use a unit circle diagram to explain why the given statement is true. $$\lim _{t \rightarrow \pi / 2} \sin t=1$$

Give an example of functions \(f\) and \(g\) and a number \(c\) such that neither \(\lim _{x \rightarrow c} f(x)\) nor \(\lim _{x \rightarrow c} g(x)\) exists, but \(\lim _{x \rightarrow c}(f(x)+g(x))\) does exist.

Use a unit circle diagram to explain why the given statement is true. $$\lim _{t \rightarrow \pi / 2} \cos t=0$$

Involve the greatest integer function \(f(x)=[| x] |\) which was defined in Example 7 on page \(145 .\) You may use your calculator as an aid in analyzing these problems. Let \(h(x)=[x]+[-x]\); find \(\lim _{x \rightarrow 2} h(x),\) if this limit exists.

Involve the greatest integer function \(f(x)=[| x] |\) which was defined in Example 7 on page \(145 .\) You may use your calculator as an aid in analyzing these problems. Let \(g(x)=x-[-x] ;\) find \(\lim _{x \rightarrow 2} g(x),\) if this limit exists.

Involve the greatest integer function \(f(x)=[| x] |\) which was defined in Example 7 on page \(145 .\) You may use your calculator as an aid in analyzing these problems. $$\text { Find } \lim _{x \rightarrow 3^{-}}(x-[x]] \text { and } \lim _{x \rightarrow 3^{+}}(x-[x])$$

Involve the greatest integer function \(f(x)=[| x] |\) which was defined in Example 7 on page \(145 .\) You may use your calculator as an aid in analyzing these problems. Let \(r(x)=\frac{[| x]|+[|-x|]}{x} ;\) find \(\lim _{x \rightarrow 3} r(x),\) if this limit exists.

Involve the greatest integer function \(f(x)=[| x] |\) which was defined in Example 7 on page \(145 .\) You may use your calculator as an aid in analyzing these problems. Let \(k(x)=\frac{x}{[| x]|+[|-x]|} ;\) find \(\lim _{x \rightarrow 1} k(x),\) if this limit exists.

If \(f(x)=\frac{1-\cos \left(x^{6}\right)}{x^{12}},\) then calculus shows that \(\lim _{x \rightarrow 0} f(x)=1 / 2 .\) A calculator or computer, however, may indicate otherwise. Graph \(f(x)\) in a viewing window with $$-.1 \leq x \leq .1$$ and use the trace feature to determine the values of \(f(x)\) when \(x\) is very close to \(0 .\) What does this suggest that the limit is?

Consider the function \(t\) whose rule is $$t(x)=\left\\{\begin{array}{ll}0 & \text { if } x \text { is rational } \\\1 & \text { if } x \text { is irrational. }\end{array}\right.$$ Explain why \(\lim _{x \rightarrow 4} t(x)\) does not exist.