91Ó°ÊÓ

Find the limit. $$\lim _{x \rightarrow \infty} \frac{\sin ^{2} x}{x^{2}}$$

\(29-30=\) Show that \(f\) is continuous on \((-\infty, \infty)\) $$f(x)=\left\\{\begin{array}{ll}{x^{2}} & {\text { if } x<1} \\ {\sqrt{x}} & {\text { if } x \geqslant 1}\end{array}\right.$$

Find the domain of the function. $$h(x)=\frac{1}{\sqrt[4]{x^{2}-5 x}}$$

(a) Estimate the value of $$\lim _{x \rightarrow 0} \frac{x}{\sqrt{1+3 x}-1}$$ by graphing the function \(f(x)=x /(\sqrt{1+3 x}-1)\) . (b) Make a table of values of \(f(x)\) for \(x\) close to 0 and guess the value of the limit. (c) Use the Limit Laws to prove that your guess is correct.

\(23-36=\) Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations. $$y=\sin \left(\frac{1}{2} x\right)$$

Find the limit. $$\lim _{x \rightarrow-\infty} \frac{1+x^{6}}{x^{4}+1}$$

\(29-30=\) Show that \(f\) is continuous on \((-\infty, \infty)\) $$f(x)=\left\\{\begin{array}{ll}{\sin x} & {\text { if } x<\pi / 4} \\ {\cos x} & {\text { if } x \geqslant \pi / 4}\end{array}\right.$$

\(23-36=\) Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations. $$y=\frac{2}{x}-2$$

Find the domain and range and sketch the graph of the function \(h(x)=\sqrt{4-x^{2}}\)

Find the numbers at which the function $$f(x)=\left\\{\begin{array}{ll}{x+2} & {\text { if } x<0} \\ {2 x^{2}} & {\text { if } 0 \leqslant x \leqslant 1} \\ {2-x} & {\text { if } x>1}\end{array}\right.$$ is discontinuous. At which of these points is \(f\) continuous from the right, from the left, or neither? Sketch the graph of \(f .\)