91Ó°ÊÓ

Find values of \(x\), if any, at which \(f\) is not continuous. $$f(x)=5 x^{4}-3 x+7$$

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

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

Complete the identities using the triangle method. $$\text { (a) } \cos \left(\tan ^{-1} x\right)=?$$ $$\text { (b) } \tan \left(\cos ^{-1} x\right)=?$$ $$\text { (c) } \sin \left(\sec ^{-1} x\right)=?$$ $$\text { (d) } \cot \left(\sec ^{-1} x\right)=?$$

Find values of \(x\), if any, at which \(f\) is not continuous. $$f(x)=\sqrt[3]{x-8}$$

Expand the logarithm in terms of sums, differences, and multiples of simpler logarithms. (a) \(\log \frac{\sqrt[3]{x+2}}{\cos 5 x}\) (b) \(\ln \sqrt{\frac{x^{2}+1}{x^{3}+5}}\)

(i) Complete the table and make a guess about the limit indicated. (ii) Confirm your conclusions about the limit by graphing a function over an appropriate interval. [Note: For the trigonometric functions, be sure to put your calculating and graphing utilities in radian mode.] $$f(x)=\frac{\cos x-1}{x^{2}} ; \lim _{x \rightarrow 0} f(x)$$ $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & -0.5 & -0.05 & -0.005 & 0.005 & 0.05 & 0.5 \\\\\hline f(x) & & & & & & \\ \hline\end{array}$$

Find the limits. $$\lim _{x \rightarrow+\infty} \sin \left(\frac{\pi x}{2-3 x}\right)$$

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 \(0<|x-a|<\delta\) $$\lim _{x \rightarrow-1 / 2} \frac{4 x^{2}-1}{2 x+1}=-2 ; \epsilon=0.05$$

Find the limits. $$\lim _{x \rightarrow-\infty} \sqrt{5-x}$$