91Ó°ÊÓ

$$\begin{aligned} &\text {a. Use a graphing utility to estimate } \lim _{x \rightarrow 0} \frac{\tan 2 x}{\sin x}, \lim _{x \rightarrow 0} \frac{\tan 3 x}{\sin x}, \text { and }\\\ &\lim _{x \rightarrow 0} \frac{\tan 4 x}{\sin x} \end{aligned}$$ b. Make a conjecture about the value of \(\lim _{x \rightarrow 0} \frac{\tan p x}{\sin x},\) for any real constant \(p\)

Evaluate each limit and justify your answer. $$\lim _{x \rightarrow 2} \sqrt{\frac{4 x+10}{2 x-2}}$$

Use analytical methods and/or a graphing utility en identify the vertical asymptotes (if any) of the following functions. $$h(x)=\frac{e^{x}}{(x+1)^{3}}$$

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. $$f(x)=1-\ln x$$

We write \(\lim _{x \rightarrow a} f(x)=-\infty\) if for any negative number \(M\) there exists \(\delta>0\) such that $$f(x)

Graph \(f(x)=\frac{\sin n x}{x},\) for \(n=1,2,3,\) and 4 (four graphs). Use the window \([-1,1] \times[0,5]\) a. Estimate \(\lim _{x \rightarrow 0} \frac{\sin x}{x}, \lim _{x \rightarrow 0} \frac{\sin 2 x}{x}, \lim _{x \rightarrow 0} \frac{\sin 3 x}{x},\) and \(\lim _{x \rightarrow 0} \frac{\sin 4 x}{x}\) b. Make a conjecture about the value of \(\lim _{x \rightarrow 0} \frac{\sin p x}{x},\) for any real constant \(p .\)

Evaluate each limit and justify your answer. $$\lim _{x \rightarrow-1}\left(x^{2}-4+\sqrt[3]{x^{2}-9}\right)$$

Use analytical methods and/or a graphing utility en identify the vertical asymptotes (if any) of the following functions. $$p(x)=\sec \left(\frac{\pi x}{2}\right), \text { for }|x|<2$$

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. $$f(x)=|\ln x|$$

Use a graphing utility to plot \(y=\frac{\sin p x}{\sin q x}\) for at least three different pairs of nonzero constants \(p\) and \(q\) of your choice. Estimate \(\lim _{x \rightarrow 0} \frac{\sin p x}{\sin q x}\) in each case. Then use your work to make a conjecture about the value of \(\lim _{x \rightarrow 0} \frac{\sin p x}{\sin q x}\) for any nonzero values of \(p\) and \(q\)