91Ó°ÊÓ

Use a graphing calculator or computer grapher to solve the equations in Exercises \(63-70 .\) $$ \sqrt{x}+\sqrt{1+x}=4 $$

Suppose that \(f(x)\) and \(g(x)\) are polynomials in \(x\) . Can the graph of \(f(x) / g(x)\) have an asymptote if \(g(x)\) is never zero? Give reasons for your answer.

Use a graphing calculator or computer grapher to solve the equations in Exercises \(63-70 .\) \(x^{3}-15 x+1=0 \quad\) (three roots)

How many horizontal asymptotes can the graph of a given rational function have? Give reasons for your answer.

Use a graphing calculator or computer grapher to solve the equations in Exercises \(63-70 .\) \(\cos x=x \quad\) (one root). Make sure you are using radian mode.

Use a graphing calculator or computer grapher to solve the equations in Exercises \(63-70 .\) \(2 \sin x=x \quad\) (three roots). Make sure you are using radian mode.

Find \(\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+x}-\sqrt{x^{2}-x}\right)\)

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?

Given \(\epsilon>0,\) find an interval \(I=(4-\delta, 4), \delta>0,\) such that if \(x\) lies in \(I,\) then \(\sqrt{4-x}<\epsilon .\) What limit is being verified and what is its value?

Greatest integer function Find (a) \(\lim _{x \rightarrow 400^{+}}\lfloor x\rfloor\) and \((b)\) \(\lim _{x \rightarrow 400^{-}}\lfloor x\rfloor ;\) then use limit definitions to verify your findings. (c) Based on your conclusions in parts (a) and (b), can anything be said about lim \(_{x \rightarrow 400}\lfloor x\rfloor ?\) Give reasons for your answers.