Last updates: 19 October 2009
1) The Fundamental Theorem of Calculus
2) Where is a Function Continuous?
3) Existence of Limits
4) Increading, Decreasing and Concave Functions
5) Evaluating Limits when
$x\to 0$
6) Evaluating Limits when
$x\to a$
7) Evaluating Limits when
$x\to \infty $
8) Limits with Exponential and Logarithm Functions
9) Limits with Trigonometric Functions
10) Limits with Inverse Trigonometric Functions
11) L'HÃ´pital's Rule
12) Sets
13) Functions
14) Ordered Sets
15) Graphs of the Basic Functions
16) Graphing Polynomials
17) Graphing Rational Functions
18) Graphing Other Functions
19) Rolle's Theorem and the Mean Value Theorem
What does $${\int}_{a}^{b}f\left(x\right)dx$$ mean?  
How does one usually calculate $${\int}_{a}^{b}f\left(x\right)dx\text{?}$$ Give an example which shows that this method does not always work. Why doesn't it?  
Give an example which shows that $${\int}_{a}^{b}f\left(x\right)dx$$ is not always the true area under $f\left(x\right)$ between $a$ and $b$ even if $f\left(x\right)$ is contunuous between $a$ and $b$.  
What is the Fundamental Theorem of Calculus?  
Let $f\left(x\right)$ be a function which is continuous and let $A\left(x\right)$ be the area under $f\left(x\right)$ from $a$ to $x$. Compute the derivative of $A\left(x\right)$ by using limits.  
Why is the Fundamental Theorem of Calculus true? Explain carefully and thoroughly.  
Give an example which illustrates the Fundamental Theorem of Calculus. In order to do this, compute an area by summing up the areas of tiny boxes and then show that applying the Fundamental Theorem of Calculus gives the same result. 
For which values of $x$ is the function $f\left(x\right)={x}^{2}+3x+4$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\frac{{x}^{2}x6}{x3}\text{,}& \text{if}x\ne 3\text{,}\\ 5\text{,}& \text{if}x=3\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\frac{\mathrm{sin}3x}{x}\text{,}& \text{if}x\ne 0\text{,}\\ 1\text{,}& \text{if}x=0\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\frac{1\mathrm{cos}x}{{x}^{2}}\text{,}& \text{if}x\ne 0\text{,}\\ 1\text{,}& \text{if}x=0\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
Determine the value of $k$ for which the function $$f\left(x\right)=\left\{\begin{array}{ll}\frac{\mathrm{sin}2x}{5x}\text{,}& \text{if}x\ne 0\text{,}\\ k\text{,}& \text{if}x=0\text{,}\end{array}\right.$$ contunuous at $x=0$. Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}x1\text{,}& \text{if}1\le x2\text{,}\\ 2x3\text{,}& \text{if}2\le x\le 3\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\mathrm{cos}x\text{,}& \text{if}x\ge 0\text{,}\\ \mathrm{cos}x\text{,}& \text{if}x0\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\mathrm{sin}\left(1/x\right)\text{,}& \text{if}x\ne 0\text{,}\\ 0\text{,}& \text{if}x=0\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
Determine the value of $a$ for which the function $$f\left(x\right)=\left\{\begin{array}{ll}ax+5\text{,}& \text{if}x\le 2\text{,}\\ x1\text{,}& \text{if}x2\text{,}\end{array}\right.$$ contunuous at $x=2$. Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}1+{x}^{2}\text{,}& \text{if}0\le x\le 1\text{,}\\ 2x\text{,}& \text{if}x1\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
For which values of $x$ is the function $f\left(x\right)=2x\leftx\right$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
Find the value of $a$ for which the function $$f\left(x\right)=\left\{\begin{array}{ll}2x1\text{,}& \text{if}x2\text{,}\\ a\text{,}& \text{if}x=2\text{,}\\ x+1\text{,}& \text{if}x2\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\frac{\leftxa\right}{xa}\text{,}& \text{if}x\ne a\text{,}\\ 1\text{,}& \text{if}x=a\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\frac{x\leftx\right}{2}\text{,}& \text{if}x\ne 0\text{,}\\ 2\text{,}& \text{if}x=0\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\mathrm{sin}x\text{,}& \text{if}x0\text{,}\\ \mathrm{x}\text{,}& \text{if}x\ge 0\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\frac{{x}^{n}1}{x1}\text{,}& \text{if}x\ne 1\text{,}\\ \mathrm{n}\text{,}& \text{if}x=1\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
Explain how you know $f\left(x\right)=\mathrm{cos}x$ is continuous for all values of $x$. Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
Explain how you know $f\left(x\right)=\mathrm{cos}\leftx\right$ is continuous for all values of $x$. Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
Explain how you know $f\left(x\right)=\u230ax\u230b$ is continuous for all values of $x$. Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
For what values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}{x}^{3}{x}^{2}+2x2\text{,}& \text{if}x\ne 1\text{,}\\ 4\text{,}& \text{if}x=1\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.  
For what values of $x$ is the function $f\left(x\right)=\leftx\right+\leftx1\right\text{,}1\le x\le 2$, contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer. 
Explain why $\underset{x\to 0}{\mathrm{lim}}\left(\frac{1}{x}\right)$ does not exist.  
Explain why $\underset{x\to \pi /2}{\mathrm{lim}}\mathrm{tan}\left(x\right)$ does not exist.  
Explain why $\underset{x\to \pi /2}{\mathrm{lim}}\mathrm{sec}\left(x\right)$ does not exist.  
Explain why $\underset{x\to 0}{\mathrm{lim}}\mathrm{csc}\left(x\right)$ does not exist.  
Explain why $\underset{x\to 1}{\mathrm{lim}}\mathrm{ln}\left(x\right)$ does not exist.  
Explain why $\underset{x\to 0}{\mathrm{lim}}\mathrm{sin}\left(\frac{1}{x}\right)$ does not exist.  
Explain why $\underset{x\to \mathrm{\infty}}{\mathrm{lim}}\mathrm{cos}\left(x\right)$ does not exist.  
Explain why
$\underset{x\to 0}{\mathrm{lim}}\mathrm{sgn}\left(x\right)$
does not exist,
where $\mathrm{sgn}\left(x\right)=\left\{\begin{array}{ll}1\text{,}& \text{if}x0\\ 0\text{,}& \text{if}x=0\\ 1\text{,}& \text{if}x0\end{array}\right.$.  
Explain why $\underset{x\to 0}{\mathrm{lim}}{2}^{1/x}$ does not exist.  
Explain why $\underset{x\to 1}{\mathrm{lim}}{2}^{1/\left(1x\right)}$ does not exist. 
What does it mean for a function $f\left(x\right)$ to be continuous at $x=a$? Explain how to test if a function is continuous at $x=a$.  
What does it mean for a function $f\left(x\right)$ to be differentiable at $x=a$? Explain how to test if a function is differentiable at $x=a$.  
What does ${\left.\left(df/dx\right)\right}_{x=a}$ indicate about the graph of $y=f\left(x\right)$? Explain why this is true. 
Evaluate $\underset{x\to 0}{lim}{\left({x}^{2}2\right)}^{2}+6$.  
Evaluate $\underset{x\to 0}{lim}\frac{5x}{x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{17x}{2x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{317x}{422x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{317x3}{422x+5}$.  
Evaluate $\underset{h\to 0}{lim}\frac{\sqrt{x+h}\sqrt{x}}{h}$.  
Evaluate $\underset{x\to 0}{lim}\frac{\sqrt{1+x+{x}^{2}}1}{x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{\sqrt{2+x}\sqrt{2}}{x}$.  
Evaluate $\underset{h\to 0}{lim}\frac{1}{h}\left(\frac{1}{\sqrt{x+h}}\frac{1}{\sqrt{x}}\right)$.  
Evaluate $\underset{x\to 0}{lim}\frac{2x}{\sqrt{a+x}\sqrt{ax}}$.  
Evaluate $\underset{x\to 0}{lim}\frac{\sqrt{1+x}1}{x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{x}{\sqrt{1+x}1}$.  
Evaluate $\underset{x\to 0}{lim}\frac{{e}^{x}+{e}^{x}2}{{x}^{2}}$.  
Evaluate $\underset{\Delta x\to 0}{lim}\frac{f\left(x+\Delta x\right)f\left(x\right)}{\left(x+\Delta x\right)x}$, when $f\left(x\right)=\sqrt{ax+b}$.  
Evaluate $\underset{\Delta x\to 0}{lim}\frac{f\left(x+\Delta x\right)f\left(x\right)}{\left(x+\Delta x\right)x}$, when $f\left(x\right)={\left(mx+c\right)}^{n}$. 
Evaluate $\underset{x\to 1}{lim}\left(6{x}^{2}4x+3\right)$.  
Evaluate $\underset{x\to 7}{lim}\frac{{x}^{2}49}{x7}$.  
Evaluate $\underset{x\to 2}{lim}\frac{{x}^{2}6x+8}{x2}$.  
Evaluate $\underset{x\to 5}{lim}\frac{2{x}^{2}+9x5}{x+5}$.  
Evaluate $\underset{x\to 1}{lim}\frac{{x}^{3}1}{x1}$.  
Evaluate $\underset{x\to 3}{lim}\frac{{x}^{2}4x+3}{{x}^{2}2x3}$.  
Evaluate $\underset{x\to 2}{lim}\frac{{x}^{3}+8}{x+2}$.  
Evaluate $\underset{x\to 3}{lim}\frac{{x}^{4}81}{x3}$.  
Evaluate $\underset{x\to 5}{lim}\frac{{x}^{5}3125}{x5}$.  
Evaluate $\underset{x\to a}{lim}\frac{{x}^{12}{a}^{12}}{xa}$.  
Evaluate $\underset{x\to a}{lim}\frac{{x}^{5/2}{a}^{5/2}}{xa}$.  
Evaluate $\underset{x\to a}{lim}\frac{{\left(x+2\right)}^{5/3}{\left(a+2\right)}^{5/3}}{xa}$.  
Evaluate $\underset{x\to 4}{lim}\frac{{x}^{3}64}{{x}^{2}16}$.  
Evaluate $\underset{x\to 2}{lim}\frac{{x}^{5}32}{{x}^{3}8}$.  
Evaluate $\underset{x\to 1}{lim}\frac{{x}^{n}1}{x1}$.  
Evaluate $\underset{x\to a}{lim}\frac{\sqrt{x}\sqrt{a}}{xa}$.  
Evaluate $\underset{x\to 2}{lim}\frac{\sqrt{3x}1}{2x}$.  
Evaluate $\underset{x\to a}{lim}\frac{\sqrt{a+2x}\sqrt{3x}}{\sqrt{3a+x}2\sqrt{x}}$.  
Evaluate $\underset{x\to a}{lim}\frac{{x}^{n}{a}^{n}}{xa}$. 
Evaluate $\underset{x\to \infty}{lim}\frac{x+2}{x2}$.  
Evaluate $\underset{x\to \infty}{lim}\frac{3{x}^{2}+2x5}{5{x}^{2}+3x+1}$.  
Evaluate $\underset{x\to \infty}{lim}\frac{{x}^{2}7x+11}{3{x}^{2}+10}$.  
Evaluate $\underset{x\to \infty}{lim}\frac{2{x}^{3}5x+7}{7{x}^{3}+{x}^{2}6}$.  
Evaluate $\underset{x\to \infty}{lim}\frac{2{x}^{3}5x+7}{7{x}^{3}+{x}^{2}6}$.  
Evaluate $\underset{x\to \infty}{lim}\frac{\left(3x1\right)\left(4x5\right)}{\left(x6\right)\left(x3\right)}$.  
Evaluate $\underset{n\to \infty}{lim}\left(\frac{1}{3}+\frac{1}{{3}^{2}}+\frac{1}{{3}^{3}}+\dots +\frac{1}{{3}^{n}}\right)$.  
Evaluate $\underset{x\to \infty}{lim}\frac{x}{\sqrt{4{x}^{2}1}1}$.  
Evaluate $\underset{x\to \infty}{lim}{2}^{x}$.  
Evaluate $\underset{n\to \infty}{lim}{\left(1+\frac{1}{n}\right)}^{n}$.  
Evaluate $\underset{t\to \infty}{lim}\frac{t+1}{{t}^{2}+1}$.  
Evaluate $\underset{n\to \infty}{lim}\sqrt{{n}^{2}+1}+n$.  
Evaluate $\underset{n\to \infty}{lim}\sqrt{{n}^{2}+n}+n$. 
Evaluate $\underset{x\to 0}{lim}\frac{{e}^{x}1}{x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{{a}^{x}1}{x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{\mathrm{ln}\left(1+x\right)}{x}$.  
Evaluate $\underset{x\to 0}{lim}{\left(1+x\right)}^{1/x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{{a}^{x}{b}^{x}}{x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{{e}^{x}+{e}^{\mathrm{x}}2}{{x}^{2}}$.  
Evaluate $\underset{x\to \infty}{lim}{2}^{x}$.  
Explain why $\underset{x\to 1}{lim}\mathrm{ln}\left(x\right)$ does not exist.  
Explain why $\underset{x\to 0}{lim}{2}^{1/x}$ does not exist.  
Explain why $\underset{x\to 1}{lim}{2}^{1/\left(x1\right)}$ does not exist.  
Evaluate $\underset{\Delta x\to 0}{lim}\frac{f\left(x+\Delta x\right)f\left(x\right)}{\left(x+\Delta x\right)x}$ where $f\left(x\right)={e}^{\sqrt{x}}$.  
Evaluate $\underset{\Delta x\to 0}{lim}\frac{f\left(x+\Delta x\right)f\left(x\right)}{\left(x+\Delta x\right)x}$ where $f\left(x\right)=ln\left(ax+b\right)$.  
Evaluate $\underset{\Delta x\to 0}{lim}\frac{f\left(x+\Delta x\right)f\left(x\right)}{\left(x+\Delta x\right)x}$ where $f\left(x\right)={x}^{x}$. 
Evaluate $\underset{x\to 0}{lim}\frac{\mathrm{sin}3x}{4x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{\mathrm{sin}x\mathrm{cos}x}{3x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{\mathrm{tan}x}{x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{1\mathrm{cos}x}{{\mathrm{sin}}^{2}x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{\mathrm{tan}ax}{\mathrm{tan}bx}$.  
Evaluate $\underset{x\to 0}{lim}\frac{\mathrm{sin}\left(x/4\right)}{x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{\mathrm{sin}mx}{\mathrm{tan}nx}$.  
Evaluate $\underset{\theta \to 0}{lim}\frac{1\mathrm{cos}6\theta}{\theta}$.  
Evaluate $\underset{x\to 0}{lim}\frac{1\mathrm{cos}2x}{3{\mathrm{tan}}^{2}x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{{\mathrm{cos}}^{2}x}{1\mathrm{sin}x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{\mathrm{tan}2xx}{3x\mathrm{sin}x}$.  
Evaluate $\underset{x\to a}{lim}\frac{\mathrm{sin}x\mathrm{sin}a}{xa}$.  
Evaluate $\underset{x\to 0}{lim}\frac{\mathrm{sin}5x\mathrm{sin}3x}{\mathrm{sin}x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{\mathrm{tan}3x2x}{3x{\mathrm{sin}}^{2}x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{{x}^{2}\mathrm{tan}2x}{\mathrm{tan}x}$.  
Evaluate $\underset{x\to \pi /4}{lim}\frac{1\mathrm{tan}x}{x\pi /4}$.  
Evaluate $\underset{x\to 0}{lim}\frac{\mathrm{tan}\left(x/2\right)}{3x}$.  
Evaluate $\underset{x\to 0}{lim}\frac{1\mathrm{cos}2x+{\mathrm{tan}}^{2}x}{x\mathrm{sin}x}$.  
Show that if $\underset{x\to 0}{lim}kx\mathrm{csc}x=\underset{x\to 0}{lim}x\mathrm{csc}kx$, then $k=\pm 1$.  
Evaluate $\underset{h\to 0}{lim}\frac{\mathrm{sin}\left(a+h\right)\mathrm{sin}a}{h}$.  
Evaluate $\underset{h\to \mathrm{\infty}}{lim}\frac{\mathrm{cos}\left(\pi /h\right)}{h2}$. 
Evaluate $\underset{x\to 1}{lim}\frac{1x}{{\mathrm{arccos}}^{2}x}$.  
Evaluate $\underset{x\to 1/\sqrt{2}}{lim}\frac{x\mathrm{cos}\left(\mathrm{arcsin}x\right)}{1\mathrm{tan}\left(\mathrm{arcsin}x\right)}$.  
Evaluate $\underset{x\to 0}{lim}\frac{x\left(1\sqrt{1{x}^{2}}\right)}{{\mathrm{arcsin}}^{3}\left(x\right)\sqrt{1{x}^{2}}}$.  
Evaluate $\underset{x\to 1}{lim}\frac{1x}{\pi 2\mathrm{arcsin}x}$.  
Evaluate $\underset{x\to 1}{lim}\frac{\mathrm{arctan}2x}{\mathrm{sin}3x}$. 
State L'HÃ´pital's rule and give an example which shows how it is used.  
Explain why L'HÃ´pital's rule works. Hint: Expand the numerator and the denominator in terms of $\Delta x$.  
Give three examples which illustrate that a limit problem that looks like it is coming out to $0/0$ could really be getting closer and closer to almost anything and must be looked at in a different way.  
Give three examples which illustrate that a limit problem that looks like it is coming out to ${1}^{\infty}$ could really be getting closer and closer to almost anything and must be looked at in a different way.  
Give three examples which illustrate that a limit problem that looks like it is coming out to ${0}^{0}$ could really be getting closer and closer to almost anything and must be looked at in a different way.  
Evaluate $\underset{x\to 1}{\mathrm{lim}}\frac{{x}^{2}+3x4}{x1}$.  
Evaluate $\underset{x\to 1}{\mathrm{lim}}\frac{{x}^{a}1}{{x}^{b}1}$.  
Evaluate $\underset{x\to 1}{\mathrm{lim}}\frac{\mathrm{ln}x}{x1}$.  
Evaluate $\underset{x\to \pi}{\mathrm{lim}}\frac{\mathrm{tan}x}{x\pi}$.  
Evaluate $\underset{x\to 3\pi /2}{\mathrm{lim}}\frac{\mathrm{cos}x}{x\left(3\pi /2\right)}$.  
Evaluate $\underset{x\to {0}^{+}}{\mathrm{lim}}\frac{\mathrm{ln}x}{\sqrt{x}}$.  
Evaluate $\underset{x\to \infty}{\mathrm{lim}}\frac{{\left(\mathrm{ln}x\right)}^{3}}{{x}^{2}}$.  
Evaluate $\underset{x\to 0}{\mathrm{lim}}\frac{{6}^{x}{2}^{x}}{x}$.  
Evaluate $\underset{x\to 0}{\mathrm{lim}}\frac{{e}^{x}1x\left({x}^{2}/2\right)}{{x}^{3}}$.  
Evaluate $\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}xx}{{x}^{3}}$.  
Evaluate $\underset{x\to \infty}{\mathrm{lim}}\frac{\mathrm{ln}\left(1+{e}^{x}\right)}{5x}$.  
Evaluate $\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{tan}\alpha x}{x}$.  
Evaluate $\underset{x\to 0}{\mathrm{lim}}\frac{2x\mathrm{arcsin}x}{2x\mathrm{arccos}x}$.  
Evaluate $\underset{x\to {0}^{+}}{\mathrm{lim}}\sqrt{x}\mathrm{ln}x$.  
Evaluate $\underset{x\to \infty}{\mathrm{lim}}{e}^{x}\mathrm{ln}x$.  
Evaluate $\underset{x\to \infty}{\mathrm{lim}}{x}^{3}{e}^{{x}^{2}}$.  
Evaluate $\underset{x\to \infty}{\mathrm{lim}}\left(x\pi \right)\mathrm{cot}x$.  
Evaluate $\underset{x\to 0}{\mathrm{lim}}{x}^{4}{x}^{2}$.  
Evaluate $\underset{x\to 0}{\mathrm{lim}}{x}^{1}\mathrm{csc}x$.  
Evaluate $\underset{x\to \infty}{\mathrm{lim}}x\sqrt{{x}^{2}1}$.  
Evaluate $\underset{x\to \infty}{\mathrm{lim}}\left(\frac{{x}^{3}}{{x}^{2}1}\frac{{x}^{3}}{{x}^{2}+1}\right)$.  
Evaluate $\underset{x\to {0}^{+}}{\mathrm{lim}}{x}^{\mathrm{sin}x}$.  
Evaluate $\underset{x\to 0}{\mathrm{lim}}{\left(12x\right)}^{1/x}$.  
Evaluate $\underset{x\to \infty}{\mathrm{lim}}{\left(1+3/x+5/{x}^{2}\right)}^{x}$.  
Evaluate $\underset{x\to \infty}{\mathrm{lim}}{x}^{1/x}$.  
Evaluate $\underset{x\to {0}^{+}}{\mathrm{lim}}{\left(\mathrm{cot}x\right)}^{\mathrm{sin}x}$.  
Evaluate $\underset{x\to \infty}{\mathrm{lim}}{\left(\frac{x}{x1}\right)}^{x}$.  
Evaluate $\underset{x\to {0}^{+}}{\mathrm{lim}}{\left(\mathrm{ln}x\right)}^{x}$. 
DeMorgan's Laws. Let
$A$,
$B$
and
$C$
be sets. Show that

Let
$S$,
$T$
and
$U$
be sets and let
$f:S\to T$
and
$g:T\to U$
be functions. Show that
 
Let $f:S\to T$ be a function and let $U\subseteq S$. The image of $U$ under $f$ is the subset of $T$ given by $$f\left(U\right)=\left\{f\left(u\right)u\in U\right\}\text{.}$$ Let $f:S\to T$ be a function. The image of $U$ under $f$ is the subset of $T$ given by $$\mathrm{im}U=\left\{f\left(s\right)s\in S\right\}\text{.}$$ Note that $\mathrm{im}f=f\left(S\right)$. Let $f:S\to T$ be a function and let $V\subseteq T$. The inverse image of $V$ under $f$ is the subset of $S$ given by $${f}^{1}\left(V\right)=\left\{s\in Sf\left(s\right)\in V\right\}\text{.}$$ Let $f:S\to T$ be a function and let $t\in T$. The fiber of $f$ over $t$ is the subset of $S$ given by $${f}^{1}\left(t\right)=\left\{s\in Sf\left(s\right)=t\right\}\text{.}$$ Let $f:S\to T$ be a function. Show that the set $F=\left\{{f}^{1}\left(t\right)t\in T\right\}$ of fibers of the map $f$ is a partition of $S$.  
 
Let $S$ be a set. The power set of $S$, ${2}^{S}$, is the set of all subsets of $S$. Let $S$ be a set and let ${\left\{0,1\right\}}^{S}$ be the set of all functions $f:S\to \left\{0,1\right\}$. Given a subset $T\subseteq S$ define a function ${f}_{T}:S\to \left\{0,1\right\}$ by $${f}_{T}\left(s\right)=\left\{\begin{array}{ll}0\text{,}& \text{if}s\notin T\text{,}\\ 1\text{,}& \text{if}s\in T\text{.}\end{array}\right.$$ Show that the map $$\begin{array}{rcll}\phi :& {2}^{S}& \u27f6& {\left\{0,1\right\}}^{S}\\ & T& \u27fc& {f}_{T}\end{array}$$ is a bijection.  
Let $\circ :S\times S\to S$ be an associative operation on a set $S$. An identity for $\circ $ is an element $e\in S$ such that $e\circ s=s\circ e=s$ for all $s\in S$. Let $e$ be an identity for an associative operation $\circ $ on a set $S$. Let $s\in S$. A left inverse for $s$ is an element $t\in S$ such that $t\circ s=e$. A right inverse for $s$ is an element $t\prime \in S$ such that $s\circ t\prime =e$. An inverse for $s$ is an element ${s}^{1}\in S$ such that ${s}^{1}\circ s=s\circ {s}^{1}=e$.
 

Show that if a greatest lower bound exists, then it is unique.  
Show that if $S$ is a lattice then the intersection of two intervals is an interval.  
A poset $S$ is left filtered if every subset $E$ of $S$ has an upper bound. A poset $S$ is right filtered if every subset $E$ of $S$ has an lower bound. Let $S$ be a poset and let $E$ be a subset of $S$. A minimal element of $E$ is an element $x\in E$ such that if $y\in E$ then $x\le y$. A poset $S$ is well ordered if every subset $E$ of $S$ has a minimal element. Show that every well ordered set is totally ordered.  
Show that there exist totally ordered sets that are not well ordered. 
Graph $f\left(x\right)=\leftx\right$.  
Graph $f\left(x\right)=\u230ax\u230b$.  
Graph $f\left(x\right)=2$.  
Graph $f\left(x\right)=x$.  
Graph $f\left(x\right)={x}^{2}$.  
Graph $f\left(x\right)={x}^{3}$.  
Graph $f\left(x\right)={x}^{4}$.  
Graph $f\left(x\right)={x}^{5}$.  
Graph $f\left(x\right)={x}^{6}$.  
Graph $f\left(x\right)={x}^{100}$.  
Graph $f\left(x\right)={x}^{1}$.  
Graph $f\left(x\right)={x}^{2}$.  
Graph $f\left(x\right)={x}^{3}$.  
Graph $f\left(x\right)={x}^{4}$.  
Graph $f\left(x\right)={x}^{100}$.  
Graph $f\left(x\right)={e}^{x}$.  
Graph $f\left(x\right)=\mathrm{sin}x$.  
Graph $f\left(x\right)=\mathrm{cos}x$.  
Graph $f\left(x\right)=\mathrm{tan}x$.  
Graph $f\left(x\right)=\mathrm{cot}x$.  
Graph $f\left(x\right)=\mathrm{sec}x$.  
Graph $f\left(x\right)=\mathrm{csc}x$.  
Graph $f\left(x\right)=\sqrt{x}$.  
Graph $f\left(x\right)={x}^{1/3}$.  
Graph $f\left(x\right)={x}^{1/4}$.  
Graph $f\left(x\right)={x}^{1/5}$.  
Graph $f\left(x\right)={x}^{1/6}$.  
Graph $f\left(x\right)=\frac{1}{\sqrt{x}}$.  
Graph $f\left(x\right)={x}^{1/3}$.  
Graph $f\left(x\right)={x}^{1/4}$.  
Graph $f\left(x\right)=\mathrm{ln}x$.  
Graph $f\left(x\right)=\mathrm{arcsin}x$.  
Graph $f\left(x\right)=\mathrm{arccos}x$.  
Graph $f\left(x\right)=\mathrm{arctan}x$.  
Graph $f\left(x\right)=\mathrm{arccot}x$.  
Graph $f\left(x\right)=\mathrm{arcsec}x$.  
Graph $f\left(x\right)=\mathrm{arccsc}x$. 
Let
$f\left(x\right)=a$,
where
$A$
is a constant.
 
Let
$f\left(x\right)=ax+b$
where
$a$
and
$b$
are constants.
 
Let
$f\left(x\right)=a\left(xc\right)+b$,
where
$a,b$
and
$c$
is a constants.
 
Let
$f\left(x\right)=\left\{\begin{array}{cc}2x\text{,}\phantom{\rule{2em}{0ex}}& \text{if}x\ge 1\text{,}\\ x\text{,}\phantom{\rule{2em}{0ex}}& \text{if}0\le x\le 1\text{.}\end{array}\right.$
 
Let
$f\left(x\right)=\left\{\begin{array}{cc}2+x\text{,}\phantom{\rule{2em}{0ex}}& \text{if}x0\text{,}\\ 2x\text{,}\phantom{\rule{2em}{0ex}}& \text{if}x\le 0\text{.}\end{array}\right.$
 
Let
$f\left(x\right)=\left\{\begin{array}{cc}1x\text{,}\phantom{\rule{2em}{0ex}}& \text{if}x1\text{,}\\ {x}^{2}1\text{,}\phantom{\rule{2em}{0ex}}& \text{if}x\ge 1\text{.}\end{array}\right.$
 
Let
$f\left(x\right)=2x{x}^{2}$.
 
Let
$f\left(x\right)=x{x}^{2}27$.
 
Let
$f\left(x\right)=3{x}^{2}2x1$.
 
Let
$f\left(x\right)={x}^{3}$.
 
Let
$f\left(x\right)={x}^{3}x+1$.
 
Let
$f\left(x\right)={x}^{3}x1$.
 
Let
$f\left(x\right)={\left(x2\right)}^{2}\left(x1\right)$.
 
Let
$f\left(x\right)=2{x}^{3}21{x}^{2}+36x20$.
 
Let
$f\left(x\right)=2{x}^{3}+{x}^{2}+20x$.
 
Let
$f\left(x\right)=1{x}^{4}$.
 
Let
$f\left(x\right)=3{x}^{4}4{x}^{3}12{x}^{2}+5$.
 
Let
$f\left(x\right)==3{x}^{4}16{x}^{3}+18{x}^{2}$.
 
Let
$f\left(x\right)={x}^{5}4{x}^{4}+4{x}^{3}$.
 
Let
$f\left(x\right)={x}^{3}{\left(x2\right)}^{2}$.
 
Let
$f\left(x\right)={\left(x2\right)}^{4}{\left(x+1\right)}^{3}\left(x1\right)$.

Let
$f\left(x\right)=y$
where
${x}^{2}+{y}^{2}=1$.
 
Let
$f\left(x\right)=\sqrt{1{x}^{2}}$.
 
Let
$f\left(x\right)=\sqrt{{a}^{2}{x}^{2}}$,
where
$a$
is a constant.
 
Let
$f\left(x\right)=y$,
where
${\left(xh\right)}^{2}+{\left(yk\right)}^{2}={r}^{2}$,
where
$h,k$
and
$r$
are constants.
 
Let
$f\left(x\right)=y$,
where
${x}^{2}+{y}^{2}2hx2ky+{h}^{2}+{k}^{2}={r}^{2}$,
where
$h,k$
and
$r$
are constants.
 
Let
$f\left(x\right)=y$
where
$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$,
where
$a$
and
$b$
are constants.
 
Let
$f\left(x\right)=y$
where
$x=a\mathrm{cos}\theta$
and
$y=b\mathrm{sin}\theta $,
where
$a$
and
$b$
are constants.
 
Let
$f\left(x\right)=\left(b/a\right)\sqrt{{a}^{2}{x}^{2}}$
where
$a$
and
$b$
are constants.
 
Let
$f\left(x\right)=y$,
where
${x}^{2}{y}^{2}=1$.
 
Let
$f\left(x\right)=y$
where
$\frac{{x}^{2}}{{a}^{2}}\frac{{y}^{2}}{{b}^{2}}=1$,
where
$a$
and
$b$
are constants.
 
Let
$f\left(x\right)=a{x}^{2}b$,
where
$a$
and
$b$
are constants.
 
Let
$f\left(x\right)=y$,
where
$x=2{y}^{2}1$.
 
Let
$f\left(x\right)=y$,
where
$x=\mathrm{cos}2\theta $
and
$y=\mathrm{cos}\theta $.
 
Let
$f\left(x\right)=b\sqrt{xa}$
where
$a$
and
$b$
are constants.
 
Let
$f\left(x\right)=\sqrt{x+2}$.
 
Let
$f\left(x\right)=\sqrt{x+2}$.
 
Let
$f\left(x\right)=y$
where
${y}^{2}\left({x}^{2}\mathrm{x}\right)={x}^{2}1$.
 
Let
$f\left(x\right)=y$
where
$x=\frac{{y}^{2}1}{{y}^{2}+1}$.
 
Let
$f\left(x\right)=\frac{\sqrt{1+x}}{\sqrt{1x}}$.
 
Let
$f\left(x\right)=\frac{{x}^{2}}{\sqrt{x+1}}$.
 
Let
$f\left(x\right)=x\sqrt{32{x}^{2}}$.
 
Let
$f\left(x\right)=x\sqrt{1{x}^{2}}$.

Let
$f\left(x\right)=\u230ax\u230b$.
 
Let
$f\left(x\right)=\leftx\right$.
 
Let
$f\left(x\right)=\leftx5\right$.
 
Let
$f\left(x\right)=\left{x}^{2}1\right$.
 
Let
$f\left(x\right)=\left\{\begin{array}{ll}1\text{,}& \text{if}x0\text{,}\\ 0\text{,}& \text{if}x=0\text{,}\\ 1\text{,}& \text{if}x0\text{.}\end{array}\right.$
 
Let
$f\left(x\right)={\left(x1\right)}^{1/3}$.
 
Let
$f\left(x\right)={x}^{2/3}$.
 
Let
$f\left(x\right)=\frac{1}{{\left(x1\right)}^{2/3}}$.
 
Let
$f\left(x\right)=x{\left(1x\right)}^{2/5}$.
 
Let
$f\left(x\right)={x}^{2/3}{\left(6x\right)}^{1/3}$.
 
Let
$f\left(x\right)=y$,
where
$\sqrt{x}+\sqrt{y}=1$.
 
Let
$f\left(x\right)=y$,
where
${x}^{2/3}+{y}^{2/3}={a}^{2/3}$,
where
$a$
is a constant.
 
Let
$f\left(x\right)=y$,
where
$x=a{\mathrm{cos}}^{3}\theta $
and
$y=a{\mathrm{sin}}^{3}\theta $.
 
Let
$f\left(x\right)=\mathrm{sin}x$.
 
Let
$f\left(x\right)=\mathrm{sin}2xx$.
 
Let
$f\left(x\right)=\mathrm{sin}x\mathrm{cos}x$,
for
$\pi /3<x<0$.
 
Let
$f\left(x\right)=2\mathrm{cos}x\mathrm{sin}2x$.
 
Let
$f\left(x\right)=\frac{\mathrm{sin}x}{x}$.
 
Let
$f\left(x\right)=\mathrm{sin}\left(1/x\right)$.
 
Let
$f\left(x\right)={e}^{x}$.
 
Let
$f\left(x\right)={e}^{1/x}$.
 
Let
$f\left(x\right)={e}^{{x}^{2}}$.
 
Let
$f\left(x\right)=\mathrm{ln}\left(4{x}^{2}\right)$.

State Rolle's theorem and draw a picture which illustrates the statement of the theorem.  
State the mean value theorem and draw a picture which illustrates the statement of the theorem.  
Explain why Rolle's theorem is a special case of the mean value theorem.  
Verify Rolle's theorem for the function $f\left(x\right)=\left(x1\right)\left(x2\right)\left(x3\right)$ on the interval $\left[1,3\right]$.  
Verify Rolle's theorem for the function $f\left(x\right)={\left(x2\right)}^{2}{\left(x3\right)}^{6}$ on the interval $\left[2,3\right]$.  
Verify Rolle's theorem for the function $f\left(x\right)=\mathrm{sin}x1$ on the interval $\left[\pi /2,5\pi /2\right]$.  
Verify Rolle's theorem for the function $f\left(x\right)={e}^{x}\mathrm{sin}x$ on the interval $\left[0,\pi \right]$.  
Verify Rolle's theorem for the function $f\left(x\right)={x}^{3}6{x}^{2}+11x6$.  
Let $f\left(x\right)=1{x}^{2/3}$. Show that $f\left(1\right)=f\left(1\right)$ but there is no number $c$ in the interval $\left[1,1\right]$ such that ${\left.\frac{df}{dx}\right}_{x=c}=0$. Why does this not contradict Rolle's theorem?  
Let $f\left(x\right)={\left(x1\right)}^{2}$. Show that $f\left(0\right)=f\left(2\right)$ but there is no number $c$ in the interval $\left[0,2\right]$ such that ${\left.\frac{df}{dx}\right}_{x=c}=0$. Why does this not contradict Rolle's theorem?  
Discuss the applicability of Rolle's theorem when $f\left(x\right)=\left(x1\right)\left(2x3\right)$ on the interval $1\le x\le 3$.  
Discuss the applicability of Rolle's theorem when $f\left(x\right)=2+{\left(x1\right)}^{2/3}$ on the interval $0\le x\le 2$.  
Discuss the applicability of Rolle's theorem when $f\left(x\right)=\u230ax\u230b$ on the interval $1\le x\le 1$.  
At what point on the curve $y=6{\left(x3\right)}^{2}$ on the interval $\left[0,6\right]$ is the tangent to the curve parallel to the $x$axis?  
Show that the equation ${x}^{5}+10x+3=0$ has exactly one real solution.  
Show that a polynomial of degree three has at most three real roots.  
Verify the mean value theorem for the function $f\left(x\right)={x}^{2/3}$ on the interval $\left[0,1\right]$.  
Verify the mean value theorem for the function $f\left(x\right)=\mathrm{ln}x$ on the interval $\left[1,e\right]$.  
Verify the mean value theorem for the function $f\left(x\right)=x$ on the interval $\left[\mathrm{a},\mathrm{b}\right]$, where $a$ and $b$ are constants.  
Verify the mean value theorem for the function $f\left(x\right)=l{x}^{2}+mx+n$ on the interval $\left[\mathrm{a},\mathrm{b}\right]$, where $l,m,n,a$ and $b$ are constants.  
Show that the mean value theorem is not applicable to the function $f\left(x\right)=\leftx\right$ in the interval $\left[1,1\right]$.  
Show that the mean value theorem is not applicable to the function $f\left(x\right)=1/x$ in the interval $\left[1,1\right]$.  
Find the points on the curve $y={x}^{3}3x$ where the tangent is parallel to the chord joining $\left(1,2\right)$ and $\left(2,2\right)$.  
If $f\left(x\right)=x\left(1\mathrm{ln}x\right)$, $x>0$, show that $\left(ab\right)\mathrm{ln}c=b\left(1\mathrm{ln}b\right)a\left(1\mathrm{ln}a\right)$, where $0<a<b$. [???] FOR SOME c IN [a,b]? 
[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561575; arXiv:math/9909077v2, MR1828302 (2002e:20083)