91Ó°ÊÓ

Given \(\epsilon \geq 0,\) 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?

Prove the limit statements. $$\lim _{x \rightarrow 0} f(x)=0 \quad \text { if } \quad f(x)=\left\\{\begin{array}{ll}2 x, & x<0 \\\x / 2, & x \geq 0\end{array}\right.$$

Use the definitions of right-hand and left-hand limits to prove the limit statements. $$\lim _{x \rightarrow 0} \frac{x}{|x|}=-1$$

Find the limits. $$\lim _{x \rightarrow(\pi / 2)^{-}} \tan x$$

Find the limits. $$\lim _{x \rightarrow-\pi} \sqrt{x+4} \cos (x+\pi)$$

Use the definitions of right-hand and left-hand limits to prove the limit statements. $$\lim _{x \rightarrow 2^{+}} \frac{x-2}{|x-2|}=1$$

\(\lim _{x \rightarrow-\pi} \sqrt{x+4} \cos (x+\pi)\) \(\lim _{x \rightarrow 0} \sqrt{7+\sec ^{2} x}\)

Find the limits. $$\lim _{x \rightarrow(-\pi / 2)^{+}} \sec x$$

Graph the function \(f\) to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function's value at \(x=0 .\) If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right or from the left? If so, what do you think the extended function's value(s) should be? $$f(x)=\frac{10|x|-1}{x}$$

Graph the function \(f\) to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function's value at \(x=0 .\) If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right or from the left? If so, what do you think the extended function's value(s) should be? $$f(x)=\frac{\sin x}{|x|}$$