Analysis

• Divide by 0 Undefined? [Kloofhs, 9/10/1996]
  When something is divided by 0, why is the answer undefined?

• Infinity over Infinity [Vigneault, 4/10/1996]
  My teacher says that infinity divided by infinity equals one. I say that infinity over infinity is indeterminate...

• Infinity, Zero [Charny, 1/4/1995]
  You can't divide by zero, but no one can actually prove WHY. . . I wanted to see a real proof first. . . We learned in trig that you can't raise zero to the zero-th power because zero would equal one, obviously. I realize infinity is not so much a number as an endless amount, but if there are an infinite number of numbers between 1 and 2, and an infinite number of numbers between 1 and 50, wouldn't the second infinity be bigger than the first?

• Line Segments and Size of Infinites [Allen, 03/19/1997]
  Divide a line segment into three parts, one half and one a quarter the length of the line segment. Choose a point at random along this line segment. What is that probability that this point lands in the 1/2 segment...?

• Patterns in Linear Systems [Chris, 11/12/2001]
  Given the equation 2x+5y=8, with the coefficients increasing by 3 each time, create linear systems that are similar using equations such as 3x+4y=5 or 4x+6y=8.

• Point nine repeating equals one? [Redding, 12/20/1994]
  When I learned how to convert repeating decimals to fractions, we were given an example in which .9 repeating equalled one. The problem I have is that I can't logically believe this is true, and I don't see an error with the math, so what am I missing or forgetting to resolve this?

• Proof of L'Hopital's Rule [Williams, 12/23/1998]
  Can you show me a proof of L'Hopital's Rule and say how it relates to the different versions of the Rule?

• Rational and Irrational Numbers [Merrill, 11/12/1997]
  Which set is bigger, the set of rational or irrational numbers?

• Rational Numbers [Oliver, 11/24/1997]
  Which is greater, the number of rational numbers between 0 and 1 or the number of rational numbers between 0 and 2?

• Finding the Difference of Two Numbers Cubed [Lee, 9/29/1995]
  The difference of two numbers is 1. The product of the two numbers is also 1. What is the difference of the 2 numbers cubed?

• Is 0.99 Periodical Equal to 1? [Koops, 4/25/1995]
  Today I had a discussion with my colleagues about whether 0.99... periodically is the same value as 1. My statement is....

• Derivation of the Power Series of Cosine [Clem, 1/22/1996]
  Would you please explain how the power series is derived from the cosine function and any other function.

• Epsilon - Delta Proofs [Miller, 1/23/1996]
  I'm confused about the f(x) in the epsilon function and the x in the delta function...

• Prove the Commutative Law [Barenboim, 4/1/1996]
  I have a question that may sound very silly: can anyone PROVE the commutative law?

• Proving/Deriving the Quadratic Equation [Mgj, 4/1/1996]
  How can one derive the quadratic equation, i.e. prove it?

• Limits and Sequencing [Ant, 5/3/1996]
  Find the limit a, and find an expression for the number N(epsilon) such that...

• Limit Evaluation [Archie, 7/7/1996]
  How do I solve the limit of the xth root of x, x -> infinity?

• Is a Ratio Rational or Irrational? [Ramnath, 8/2/1996]
  Is there any way by which we could ascertain the irrationality of a ratio (eg. 4/1, 22/7) without actually resorting to division?

• Meaning of R++ [Cook, 9/12/1996]
  What do the symbols R_{+} and R_{++} mean?

• Real Analysis [Moore, 9/13/1996]
  Let I be a subset of R where I is an interval and let f map I to R be increasing on I...

• Teaching Synthetic Division [Kenyon, 10/23/1996]
  Do you have any advice for teaching synthetic division, on the computer, to the rest of my class?

• Sizes of Infinities [Clackum, 01/31/1997]
  How can you prove that one infinity is larger than another?

• Math Analysis [Raymond, 09/13/1997]
  I have to figure out the boundaries for the inequality...

• Indeterminate Forms [White, 09/18/1997]
  What is infinity divided by infinity?

• Prove That a Set Is Uncountably Infinite [Ramos, 10/31/1997]
  How can one prove that the set [0,1]x[0,1] is uncountably infinite?

• Alternating Harmonic Series [Campbell, 11/18/1997]
  I am trying to find the proof for the sum of the alternating harmonic series. I did find out that it is ln(2), but please tell me why?

• Eternity [Sibilang, 12/30/1997]
  If you must choose from two alternatives, eternity or half of eternity, which is the best?

• Factoring Quadratics [Stanfield, 03/05/1998]
  Factoring quadratics that occur in fractions.

• Closure and the Reals [Zobell, 03/26/1998]
  Under what set of operations are the positive real integers closed?

• Two Definitions of Limits, with Examples [Cain, 05/11/1998]
  Epsilon-delta definitions of the limit of a function and the limit of a sequence.

• A Limit Proof Using Estimation [Mostyn, 06/01/1998]
  How do I show that the limit as of x^2 x->(-2) is 4, using the delta-epsilon definition of a limit?

• Convergent and Divergent Series [Gena, 07/14/1998]
  I cannot seem to solve these problems...

• Limits of the Natural Logarithm [Andrew, 07/22/1998]
  Can you help us find the limit of x (ln x)^n as x goes to infinity, for all n? Do we use L'Hopital's Rule?

• Equations with Infinity [Rundle, 07/29/1998]
  How can you use infinity in maths equations? Is it like other numbers?

• Formal Definition of a Limit [Sabroski, 08/01/1998]
  What is the connection between the open intervals and the informal definition of the limit?

• Descartes' Rule of Signs and Complex Roots [McDowell, 08/04/1998]
  Prove that if p and q are real and q is not equal to 0, the equation x^3 + px +q = 0 has two imaginary roots.

• The Impossibility of Integrating x^x [Black, 08/07/1998]
  How would you test to see if the integral of x^x can be expressed in a finite number of elementary functions?

• Limit Problems [Josephine, 08/08/1998]
  What is the limit of (sqrt(2-t) - sqrt(2))/t as t->0?

• Fourier Series and the Zeta Function [Perry, 08/13/1998]
  How do you evaluate Zeta of 2?

• Limit Proofs [Donovan, 08/23/1998]
  Show that the limit as x approaches 0 of 1/x-1 = -1.

• Delta-Epsilon Limit Proofs [Angeles, 09/05/1998]
  Can you help me with some delta-epsilon proofs? How do you choose delta?

• Countability [Stephanie, 09/09/1998]
  How can I show that the rationals are countable and the irrationals are uncountable?

• The Intermediate Value Theorem [Book, 09/14/1998]
  Can you explain the Intermediate Value Theorem? How does it fail with this example: f(x) = 1/(x-3) + 2?

• An Uncountable Set [Celenti, 09/28/1998]
  How do you prove, using diagonalization, that the set of functions from N to N is uncountable?

• Limit Intuitions [Bryan, 10/01/1998]
  Can you explain the intuition behind the formal definition of a limit? derive the equation of the tangent line of a function at a given point?

• A Sorting Lower Bound [Smither, 10/01/1998]
  Show that you cannot sort 3 elements with 2 comparisons, an example of lower bound.

• The Monotone Convergence Theorem [Flarity, 10/02/1998]
  Could you please explain the meaning and purpose of the monotone convergence theorem?

• The Limit of Sin(1/x) [Johnson, 10/02/1998]
  Why does the sine of 1/x have no limit as x approaches 0?

• Open Balls and Open Sets [Robert, 10/04/1998]
  Can you explain the concepts of open balls and open sets? What about in the plane (2D) and in space (3D)?

• Infinity Minus Infinity [Kuster, 10/23/1998]
  What would you get if you subtracted infinity from infinity? Would it not be the same as x minus x?

• Onto (Surjective) Functions [Taro, 11/03/1998]
  I don't understand onto functions. Would you please explain them in detail?

• What is 1^infinity? [Hiscock, 12/10/1998]
  Why is 1^infinity indeterminate? How can you show this using limits? Why is it not equal to 1, as intuition leads us to believe?

• Proving Limits Formally [Glover, 01/04/1999]
  Given a sequence, I can guess what the limit it. How do I actually prove my guess is the limit?

• Riemann Sums and the Integral [Brian, 01/18/1999]
  Why does the limit of Riemann sums turn out to be the integral? How is that derived? What is behind it?

• Using L'Hopital's Rule [Mirza, 01/22/1999]
  Prove that e^(-1/3) is the limit for y =(arctan(x)/x)^(1/(x^2)) when x -> 0.

• Singularity [Eric, 03/01/1999]
  What is a singularity?

• Rational/Irrational Numbers [Jim, 03/12/1999]
  Prove that between any two numbers there are a rational and an irrational number and, therefore, that there are infinite numbers of each.

• Fractions in Ascending Order [David, 03/22/1999]
  Rearrange b/a, d/c, bd/ac, b+d/a+c, 1 in ascending order...

• Natural Logarithm and Limits [Johnson, 03/25/1999]
  Prove that lim (ln x) = ln (lim x] .

• Nonoverlapping Intervals [Cheng, 04/08/1999]
  Show that there can not be an uncountable number of nonoverlapping intervals on the real line.

• Mean Value Theorem [Porter, 04/18/1999]
  Which number satisfies the conclusion of the Mean Value Theorem for f(x) = sin (x/2)?

• Dense and Nowhere Dense Sets [Patrick, 04/25/1999]
  Can you define the mathematical terms 'dense' and 'nowhere dense'?

• Convergence of Sums [Andrews, 05/07/1999]
  Deduce that the following sum converges absolutely: Sum from 1 to infinity of (-1)^(n-1)/(n^2)!

• Significant Difference? [Rita, 05/08/1999]
  A research question: does mean delivery time differ for boy babies and girl babies?

• Fibonacci Sequence - An Example [Chow, 05/12/1999]
  Glass plates and reflections.

• Passing Ships [Covington, 05/12/1999]
  Every day a ship leaves San Francisco for Tokyo... How many Tokyo ships will each San Francisco ship meet?

• Using Vectors to Prove Collinearity [Glenn, 06/05/1999]
  How can I prove that the points of intersection of AB.DE, BC.EF, and CA.FD are collinear, if concurrent lines 0A, 0B, and 0C are produced to D, E, and F?

• Phi Function [Hopkins, 07/07/1999]
  Given n and m as subjects, does phi(n)*phi(m) = phi(n*m)?

• Perimeter of 1000m [Macdonald, 07/13/1999]
  Find the shape with perimeter 1000m and the largest possible area.

• Fixed Point of an Equation [Jharon, 07/21/1999]
  Find the fixed point of T(x)=1/x^2.

• Sum of Convergent Series [Abby, 09/25/1999]
  How can you find the sum for k = 0 to infinity of 1/[(k+1)(k+3)], and the sum for k = 0 to infinity of [(25/10^k) - (6/100^k)]?

• What is 0.999... + 0.999...? [Jarman, 09/27/1999]
  How can I show that 0.999... + 0.999... = 2?

• Complex Numbers in Quadratic Equations [Oya, 11/09/1999]
  How are imaginary numbers used in solving quadratic equations? How can solutions of this type be represented graphically?

• Minimizing the Length of a Crease [Johnston, 11/11/1999]
  How can I minimize the the length (L) of the crease if one of the corners of an 8.5" x 11" sheet of paper is lifted up and placed on the opposite longer edge, and then the paper is smoothed flat?

• Proving the Properties of Natural Numbers [Harsha, 03/08/2000]
  How can you prove or derive the commutative, associative, and distributive properties of numbers?

• Induction on .999... [Thomson, 10/19/2000]
  In the FAQ proof that .999... = 1, how can you multiply .999... by 10 if you can never get to the furthest right value? Can you show me an induction proof that this works?

• Analysis and the Derivative [Yvonne, 01/23/2001]
  Suppose that f:RtoR is differentiable at c and that f(c)=0. Show that g(x):=|f(x)| is differentiable at c if and only if f'(c)=0.

• Indeterminate Forms [Brian, 04/23/2001]
  Concerning the indeterminate forms such as 0/0 and infinity/infinity, why is one to the infinite power considered an indeterminate form?

• Creating a Smooth Acceleration in Animation [William, 05/09/2001]
  I am trying to figure out a way to set the value of X to achieve a specific difference between the last two sets in this problem...

• Continuity [Cathy, 11/24/2001]
  Suppose f and g are continuous on I = [a,b]... Prove there exists a k greater than 0 such that f(x) + k is less than or equal to g(x) for all x in I.

• Proofs of e [Christine, 03/21/2002]
  I found out that the definition of e is: e = lim, as n approaches infinity (1+1/n)^n. Is there a proof for this or for 1+2(e-1/n)?