91影视

For each of the following sign charts, sketch the graph of a function f that has the indicated signs, zeros, and discontinuities:

Find the equation of a rational function that could have the graph shown. Take into account roots, holes, and vertical and horizontal asymptotes when constructing your function.

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.

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

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

for sufficiently small negative angles 胃.

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.

$\underset{x鈫�1^{-}}{\lim }\sqrt{1-x}=0$

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

$\underset{x鈫�2}{lim}鈥�f\left(x\right)=鈭�4\text{and}\underset{x鈫�鈭�\mathrm{鈭�}}{lim}鈥�f\left(x\right)=鈭�\mathrm{鈭�}$

Suppose $f\left(x\right)=mx+b$is a linear function with $m鈮�0$, and let c be any real member.

Part (a): Show that for all $鈭�>0$, if $0<\left|x-c\right|<\frac{鈭�}{\left|m\right|}$, then $\left|f\left(x\right)-f\left(c\right)\right|<鈭�$.

Part (b): What does the implication in part (a) have to do with limits?

Part (c): Illustrate the implication in part (a) with a labeled graph. Explain in terms of slopes why it makes sense that given $鈭�>0$, the corresponding$未>0$is$未=\frac{鈭�}{\left|m\right|}$.

Find the limit by hand.

$\underset{x鈫�-鈭�}{\lim }{e}^x{\tan }^{-1}x$

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$0<\left|x-2\right|<未$, then$\left|\left(3x-1\right)-5\right|<0.25$.

Write each limit in Exercises 19鈥�42 as a formal statement involving $未,蔚,N,\mathrm{and}/\mathrm{or}M,$v 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.

$\underset{x鈫�2}{\lim }\frac{x^2-4}{x-2}=4$