91Ó°ÊÓ

Find the exact value of the given quantity. $$\sin \left[2 \cos ^{-1}\left(\frac{3}{3}\right)\right]$$

A positive number \(\epsilon\) and the limit \(L\) of a function \(f\) at \(a\) are given. Find a number \(\delta\) such that \(|f(x)-L|<\epsilon\) if \(0<|x-a|<\delta\) $$\lim _{x \rightarrow 4} 2 x=8 ; \epsilon=0.1$$

In each part determine whether the function is continuous or not, and explain your reasoning. (a) The Earth's population as a function of time. (b) Your exact height as a function of time. (c) The cost of a taxi ride in your city as a function of the distance traveled. (d) The volume of a melting ice cube as a function of time.

A positive number \(\epsilon\) and the limit \(L\) of a function \(f\) at \(a\) are given. Find a number \(\delta\) such that \(|f(x)-L|<\epsilon\) if \(0<|x-a|<\delta\) $$\lim _{x \rightarrow 3}(5 x-2)=13 ; \epsilon=0.01$$

Find the limits. $$\lim _{x \rightarrow+\infty}\left(2 x^{3}-100 x+5\right)$$

Find the limits. $$\lim _{x \rightarrow 2} \frac{x^{2}-4 x+4}{x^{2}+x-6}$$

Find the limits. $$\lim _{x \rightarrow+\infty} \cos \left(\frac{1}{x}\right)$$

Expand the logarithm in terms of sums, differences, and multiples of simpler logarithms. (a) \(\log (10 x \sqrt{x-3})\) (b) \(\ln \frac{x^{2} \sin ^{3} x}{\sqrt{x^{2}+1}}\)

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

Find the limits. $$\lim _{x \rightarrow+\infty} \sqrt{x}$$