91Ó°ÊÓ

Read the section and make your own summary of the material.

Read the section and make your own sum-

mary of the material.

Problem Zero: Read the section and make your own summary of the material.

Read the sections and make your own summary of the material.

Read the section and make your own summary of material.

Read the section and make your own summary of the material.

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) A limit exists if there is some real number that it is equal to.

(b) The limit of $f\left(x\right)$as $x→c$is the value $f\left(c\right)$.

(c) The limit of $f\left(x\right)$as $x→c$might exist even if the value of $f\left(c\right)$does not.

(d) The two-sided limit of $f\left(x\right)$as $x→c$exists if and only if the left and right limits of $f\left(x\right)$exists as $x→c$.

(e) If the graph of $f$has a vertical asymptote at $x=5$, then $\underset{x→5}{\lim }f\left(x\right)=∞$.

(f) If $\underset{x→5}{\lim }f\left(x\right)=∞$, then the graph of $f$has a vertical asymptote at $x=5$.

(g) If $\underset{x→2}{\lim }f\left(x\right)=∞$, then the graph of $f$has a horizontal asymptote at $x=2$.

(h) If$\underset{x→∞}{\lim }f\left(x\right)=2$, then the graph of$f$has a horizontal asymptote at$y=2$.

Explain why it makes intuitive sense that$\underset{x→c}{lim} x=c$ for any real number c. Then use a delta–epsilon argument to prove it.

Finding roots of piecewise-defined functions: For each function f that follows, find all values x = c for which f(c) = 0. Check your answers by sketching a graph of f.

$\begin{array}{r}f\left(x\right)=\left\{\begin{array}{rl}4−x^2,& \text{if}x<0\\ x+1,& \text{if}xâ‰\yen 0\end{array}\right.\\ f\left(x\right)=\left\{\begin{array}{rl}x+1,& \text{if}x<0\\ 4−x^2,& \text{if}xâ‰\yen 0\end{array}\right.\\ f\left(x\right)=\left\{\begin{array}{rl}2x−1,& \text{if}x≤1\\ 2x^2+x−3,& \text{if}x>1\end{array}\right.\end{array}$

For each sequence shown, find the next two terms. Then write a general form for the kth term of the sequence.

$$\begin{gathered} 2,6,10,14,18,22,… \\ 1,8,27,64,125,216,… \\ 3,\frac{3}{4},\frac{3}{9},\frac{3}{16},\frac{3}{25},\frac{3}{36},... \\ 1,\frac{1}{3},\frac{1}{9},\frac{1}{27},\frac{1}{81},\frac{1}{243},… \\ \frac{3}{5},\frac{4}{7},\frac{5}{9},\frac{6}{11},\frac{7}{13},\frac{8}{15},… \\ \frac{3}{2},\frac{5}{5},\frac{7}{10},\frac{9}{17},\frac{11}{26},\frac{13}{37},… \end{gathered}$$