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High School Archive

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Fixed Point of an Equation [07/21/1999]

Find the fixed point of T(x)=1/x^2.

Fixed-Point Theorem [05/02/2002]

An intuitive reason why the theorem is true, and a proof.

Formal Definition of a Limit [08/01/1998]

What is the connection between the open intervals and the informal definition of the limit?

Formulas for N-Dimensional Spheres [10/26/2000]

What is the next term in the sequence pi*r^2, (4/3)pi*r^3, ...? Would it be the formula for the volume of a 4-dimensional sphere?

Fourier Series and the Zeta Function [08/13/1998]

How do you evaluate Zeta of 2?

The Impossibility of Integrating x^x [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?

Indeterminate Forms [09/18/1997]

What is infinity divided by infinity?

Indeterminate Forms [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?

Induction on .999... [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?

Infinite Series Involving Arithmetic and Geometric Means [11/27/2003]

Start with two numbers, find their arithmetic and geometric means, then find the means of the two results and continue this process indefinitely. Does the series converge?

Infinity Minus Infinity [10/23/1998]

What would you get if you subtracted infinity from infinity? Would it not be the same as x minus x?

The Intermediate Value Theorem [09/14/1998]

Can you explain the Intermediate Value Theorem? How does it fail with this example: f(x) = 1/(x-3) + 2?

Intermediate Value Theorem [09/19/2002]

How can we prove by the intermediate value theorem that there is a point on the path that a hiker will cross at exactly the same time of the day hiking up and returning?

Is 0.99 Periodical Equal to 1? [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....

Is a Ratio Rational or Irrational? [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?

Is the Number Line Both Continuous and Porous? [05/13/2005]

The number line is said to be densely populated because between any two numbers are infinite other numbers. But aren't there also an infinite number of gaps in the number line, with a tiny gap on either side of each of those discrete infinite numbers?

Leftmost Digits of 2^n [11/01/2002]

Prove that there is a power of 2 whose decimal representation starts with the digits 1999.

Limit Evaluation [7/7/1996]

How do I solve the limit of the xth root of x, x -> infinity?

Limit Intuitions [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?

The Limit of Sin(1/x) [10/02/1998]

Why does the sine of 1/x have no limit as x approaches 0?

Limit of x sin(1/x) [04/23/2002]

I assumed from the graph that the function had a limit at x=0 of 0, but since it involves sin(1/0) I can not prove this using the basic trigonometric limits (sin x/x and (1-cos x)/x), L'Hopital's rule, or by rearranging the equation. Can you help?

Limit Problems [08/08/1998]

What is the limit of (sqrt(2-t) - sqrt(2))/t as t->0?

Limit Proofs [08/23/1998]

Show that the limit as x approaches 0 of 1/x-1 = -1.

A Limit Proof Using Estimation [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?

Limits and Sequencing [5/3/1996]

Find the limit a, and find an expression for the number N(epsilon) such that...

Limits of the Natural Logarithm [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?

Math Analysis [09/13/1997]

I have to figure out the boundaries for the inequality...

Meaning of R++ [9/12/1996]

What do the symbols R_{+} and R_{++} mean?

Mean Value Theorem [04/18/1999]

Which number satisfies the conclusion of the Mean Value Theorem for f(x) = sin (x/2)?

Minimizing the Length of a Crease [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?

The Monotone Convergence Theorem [10/02/1998]

Could you please explain the meaning and purpose of the monotone convergence theorem?

Natural Logarithm and Limits [03/25/1999]

Prove that lim (ln x) = ln (lim x] .

Nonoverlapping Intervals [04/08/1999]

Show that there can not be an uncountable number of nonoverlapping intervals on the real line.

Onto (Surjective) Functions [11/03/1998]

I don't understand onto functions. Would you please explain them in detail?

Open Balls and Open Sets [10/04/1998]

Can you explain the concepts of open balls and open sets? What about in the plane (2D) and in space (3D)?

Passing Ships [05/12/1999]

Every day a ship leaves San Francisco for Tokyo... How many Tokyo ships will each San Francisco ship meet?

Perimeter of 1000m [07/13/1999]

Find the shape with a perimeter of 1000m and the largest possible area.

Phi Function [07/07/1999]

Given n and m as subjects, does phi(n)*phi(m) = phi(n*m)?

Product of Negative Numbers [02/04/1997]

Can you give a proof of why the product of two negative numbers is positive?

Proof Involving Rational and Irrational Numbers [09/13/2004]

Prove the existence of 2 irrational numbers, a and b, where a^b is rational.

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