91影视

Write down the formal delta-epsilon statement you would have to prove in order to prove the limit statement.

$\underset{x鈫�-2}{lim}\frac{3}{x+1}=-3$

Fill in the blanks to complete each of the following theorem statements:

For $未>0,x鈭�(c-未,c)鈭�(c,c+未)$if and only if $0<\underline{}<未.$

Calculate the derivative of $f\left(x\right)=x^2$at $c=4$.

Interesting trigonometric limits: For each of the functions that follow, use a calculator or other graphing utility to examine the graph of f near x = 0. Does it appear that f is continuous at x = 0? Make sure your calculator is set to radian mode

$f\left(x\right)=\left\{\begin{array}{r}x\sin 鈦�\left(\frac{1}{x}\right),\text{if}x鈮�0\\ 0,\text{if}x=0\end{array}\right.$

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that k is positive

$\underset{x鈫�c}{\lim }(mx+b)=?$

State the constant multiple rule, sum rule, product rule, quotient rule, and composition rule for limits.

Calculating limits: Find each limit by hand.

$\underset{x鈫�2}{\lim }\frac{1}{2-x}.$

If $f$is a continuous function, what can you say about$\underset{x鈫�1}{\lim }f\left(x\right)?$

Use the preceding two problems and the result of Exercise 22 to calculate the following limit

$\begin{array}{l}\underset{x鈫�鈭�1}{lim}鈥�x鈥呪赌呪赌呪赌�鈰�\underset{x鈫�4}{lim}鈥�x鈥呪赌呪赌呪赌�\underset{x鈫�蟺}{lim}鈥�x\\ \underset{x鈫�0}{lim}鈥�x^2鈥呪赌呪赌呪赌�鈰�\underset{x鈫�5}{lim}鈥�x^2{x}_{x鈫�鈭�2}\\ \underset{x鈫�0}{lim}鈥�(2x鈭�3)鈥呪赌呪赌呪赌�鈰�\underset{x鈫�1}{lim}鈥�(1鈭�x)鈥呪赌呪赌呪赌�鈰�\underset{x鈫�3}{lim}鈥�(3x+1)\end{array}$

A limit representing an instantaneous rate of change: After t seconds, a bowling ball dropped from $350$feet has height $h\left(t\right)=350-16t^2,$measured in feet.

Take the limit as $h鈫�0^{+}$of the formula you found for average rate of change in the previous problem. What does this limit represent in real-world terms?