91影视

Find the limit by hand.

$\underset{x鈫�0}{\lim }\frac{3x}{\sin 2x}$

Find the roots, discontinuous, and horizontal and vertical asymptotes of the function in Exercises 23-24. Support your answers by explicitly computing any relevant limits.

$f\left(x\right)=\frac{x^2-2x-3}{x-3}$

Limits of combinations: Fill in the blanks to complete the limit

rules that follow. You may assume that k and c are any real

numbers and that both $\underset{x鈫�c}{\lim }f\left(x\right)\mathrm{and}\underset{\mathrm{x}鈫�\mathrm{c}}{\lim }g\left(\mathrm{x}\right)$exist.

$\underset{x鈫�c}{\lim }\left(f\left(x\right)-g\left(x\right)\right)=\_\_\_\_\_\_$

Sketch the graphs of functions that have the given limits and values. (There are multiple correct answers.)

$\underset{x鈫�0^{+}}{lim}鈥�f\left(x\right)=\mathrm{鈭�}\text{and}\underset{x鈫�0^{鈭�}}{lim}鈥�f\left(x\right)=\mathrm{鈭�}$

For each function f graphed in Exercises 23鈥�26, describe the intervals on which f is continuous. For each discontinuity of f, describe the type of discontinuity and any one-sided continuity. Justify your answers about discontinuities with limit statements.

Use a geometric argument and the Squeeze Theorem for limits to argue that $\underset{胃鈫�0^{鈭�}}{lim}鈥�\cos 鈦�胃=1$

for sufficiently small negative angles 胃.

Use algebra to find the largest possible value of 未 or smallest possible value of N that makes each implication is true. Then verify and support your answers with labeled graphs.

If $0<\left|x-3\right|<未$, then$\left|\frac{1}{x}-\frac{1}{3}\right|<0.2$

In this exercise you will use a calculator to investigate the number e.

1. Make a table of values that describes the behavior of the quantity $(1+h{)}^{1/h}$as $h鈫�0$.
2. Make a table of values that describes the behavior of the quantity $\frac{e^h-1}{h}$as $h鈫�0$.
3. What do your tables of values have to do with Definition 1.25 and Theorem 1.26?

Write each limit in Exercises 19鈥�42 as a formal statement involving $未,蔚,N,\mathrm{and}/\mathrm{or}M,$and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

localid="1648643296908" $\underset{x鈫�1}{\lim }\left(x^2-3\right)=-2$

Limits of combinations: Fill in the blanks to complete the limit

rules that follow. You may assume that k and c are any real

numbers and that both $\underset{x鈫�c}{\lim }f\left(x\right)\mathrm{and}\underset{\mathrm{x}鈫�\mathrm{c}}{\lim }g\left(\mathrm{x}\right)$exist.

$\underset{x鈫�c}{\lim }\left(f\left(x\right)g\left(x\right)\right)=\_\_\_\_\_\_$