From a9ebb42059647ccb91efd4eb827578b29b154119 Mon Sep 17 00:00:00 2001 From: Marica Odagaki Date: Thu, 3 Apr 2014 23:39:14 +0900 Subject: [PATCH] Fix broken HTML in limitCompute exercise --- .../exercises/limitCompute.html | 454 ------------------ 1 file changed, 454 deletions(-) diff --git a/public/khan-exercises/exercises/limitCompute.html b/public/khan-exercises/exercises/limitCompute.html index b82909c..5e54602 100644 --- a/public/khan-exercises/exercises/limitCompute.html +++ b/public/khan-exercises/exercises/limitCompute.html @@ -36,90 +36,6 @@ D*(A-B)/(E*(A-C)) -
-

Note we cannot evaluate this limit directly since the - denominator is zero when x=A.

- -

Start by factoring the numerator and the - denominator.

- -

In this case we find: -

- - \displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()} = - \lim_{x \to A}\frac{(lin2)(lin1)}{(lin3)(lin1)} - -

- -

Since we are taking the limit as x\to A, we may assume that x\ne A. -

- -

Canceling factors we see: -

- - \displaystyle\lim_{x \to A}\frac{(lin2)(lin1)}{(lin3)(lin1)}= - \lim_{x \to A}\frac{(lin2)}{(lin3)} - -

- -

We can evaluate this rational function at x = A. -

- -

Hence, -

- - \displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()} = - \frac{D(A) +-B*D}{E(A)+-C*E}. - -

-

Finally, -

- - \displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()} = - \dfrac{toFraction(D*(A-B)/(E*(A-C)))[0]}{toFraction(D*(A-B)/(E*(A-C)))[1]}. - -

-
- - - - - - - Computing Limits - - - -
-
-
- -
- randRangeNonZero(-9,9) - randRangeExclude(-9,9,[A]) - randRangeExclude(-9,9,[A,B]) - randRangeNonZero(-9,9) - randRangeNonZero(-9,9) - new Polynomial(0,1,[-A,1]) - new Polynomial(0,1,[-B*D,D]) - new Polynomial(0,1,[-C*E,E]) - lin1.multiply(lin2) - lin1.multiply(lin3) - \frac{num.text()}{denom.text()} -
- -

- Evaluate the following limit: -

- - \displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()} - -

- -
- D*(A-B)/(E*(A-C)) -
-

Note we cannot evaluate this limit directly since the denominator is zero when x=A.

@@ -535,374 +451,4 @@
- - -
- -
- randRangeNonZero(-9,9) - randRangeExclude(-9,9,[A]) - randRangeExclude(-9,9,[A,B]) - randRangeNonZero(-9,9) - randRangeNonZero(-9,9) - new Polynomial(0,1,[-A,1]) - new Polynomial(0,1,[-B*D,D]) - new Polynomial(0,1,[-C*E,E]) - lin1.multiply(lin2) - lin1.multiply(lin3) - \frac{num.text()}{denom.text()} -
- -

- - Evaluate the following limit: -

- - \displaystyle\lim_{x \to A}\frac{num.text()}{lin1.text()} - -

- -
- D*A-B*D -
- -
-

Note we cannot evaluate this limit directly since the - denominator is zero when x=A.

- -

Start by factoring the numerator.

- -

In this case we find: -

- - \displaystyle\lim_{x \to A}\frac{num.text()}{lin1} = - \lim_{x \to A}\frac{(lin2)(lin1)}{lin1} - -

- -

Since we are taking the limit as x\to A, we may assume that x\ne A. -

- -

Canceling factors we see: -

- - \displaystyle\lim_{x \to A}\frac{(lin2)(lin1)}{(lin3)(lin1)}= - \lim_{x \to A}(lin2) - -

- -

We can evaluate this linear function at x = A. -

- -

Hence, -

- - \displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()} = - D(A) +-B*D. - -

-

- Finally, we see the solution is D*(A-B). -

-
- -
- -
-
- randRangeNonZero(-9,9) -
- -

- Evaluate the following limit: -

- - \displaystyle\lim_{x \to A}\frac{x^2-A*A}{x-A} - -

- -
- 2*A -
- -
-

Note we cannot evaluate this limit directly since the - denominator is zero when x=A.

- -

Start by factoring the numerator.

- -

In this case we find: -

- - \displaystyle\lim_{x \to A}\frac{x^2-A*A}{x-A} = - \lim_{x \to A}\frac{(x+A)(x-A)}{x-A} - -

- -

Since we are taking the limit as x\to A, we may assume that x\ne A. -

- -

Canceling factors we see: -

- - \displaystyle\lim_{x \to A}\frac{(x+A)(x-A)}{x-A} = - \lim_{x \to A}(x+A) - -

- -

We can evaluate this linear function at x = A. -

- -

Hence, -

- - \displaystyle\lim_{x \to A}(x+A) = - A +A. - -

-

Finally, we see the solution is -

- - 2*A. - -

-
- -
- -
-
- randRangeNonZero(-9,9) -
-

- Evaluate the following limit: -

- - \displaystyle\lim_{x \to A}\frac{x^3-A*A*A}{x-A} - -

- -
- 3*A*A -
- -
-

Note we cannot evaluate this limit directly since the - denominator is zero when x=A.

- -

Start by factoring the numerator.

- -

In this case we find: -

- - \displaystyle\lim_{x \to A}\frac{x^3-A*A*A}{x-A} = - \lim_{x \to A}\frac{(x^2+Ax +A*A)(x-A)}{x-A} - -

- -

Since we are taking the limit as x\to A, we may assume that x\ne A. -

- -

Canceling factors we see: -

- - \displaystyle\lim_{x \to A}\frac{(x^2+Ax +A*A)(x-A)}{x-A} = - \lim_{x \to A}(x^2+Ax +A*A) - -

- -

We can evaluate this quadratic function at x = A. -

- -

Hence, -

- - \displaystyle\lim_{x \to A}(x^2+Ax +A*A) = - (A)^2 +A(A) + A*A. - -

-

Finally, we see the solution is -

- - 3*A*A. - -

-
- -
- -
-
- randRangeNonZero(-9,9) -
- -

-

- Evaluate the following limit: -

- - \displaystyle\lim_{x \to A}\frac{x^4-A*A*A*A}{x-A} - -

- -
- 4*A*A*A -
- -
-

Note we cannot evaluate this limit directly since the - denominator is zero when x=A.

- -

Start by factoring the numerator.

- -

In this case we find: -

- - \displaystyle\lim_{x \to A}\frac{x^4-A*A*A*A}{x-A} = - \lim_{x \to A}\frac{(x^3+Ax^2 +A*Ax + A*A*A)(x-A)}{x-A} - -

- -

Since we are taking the limit as x\to A, we may assume that x\ne A. -

- -

Canceling factors we see: -

- - \displaystyle\lim_{x \to A}\frac{(x^3+Ax^2 +A*Ax + A*A*A)(x-A)}{x-A}= - \lim_{x \to A}(x^3+Ax^2 +A*Ax + A*A*A) - -

- -

We can evaluate this cubic function at x = A. -

- -

Hence, -

- - \displaystyle\lim_{x \to A}(x^3+Ax^2 +A*Ax + A*A*A)= - (A)^3+A(A)^2 +A*A(A) + A*A*A. - -

-

Finally, we see the solution is -

- - 4*A*A*A. - -

-
- -
- -
-
- randRangeNonZero(-9,9) - randRangeExclude(-9,9,[0,A]) -
- -

- Evaluate the following limit: -

- - \displaystyle\lim_{x \to 0}\frac{\sin(Ax)}{Bx} - -

- -
- A/B -
- -
-

Recall that -

- - \displaystyle\lim_{x\to 0}\frac{\sin(x)}{x} = 1. - -

-

Factor out \frac{B}{A} in the denominator to get -

- \displaystyle\lim_{x \to 0}\frac{\sin(Ax)}{Ax \cdot \frac{B}{A}} - = \lim_{x \to 0}\frac{\sin(Ax)}{Ax} \cdot \frac{A}{B} - -

- -

- Set y = Ax. Since y\to 0 as x\to 0, -

- \displaystyle\lim_{x \to 0}\frac{\sin(Ax)}{Ax} \cdot \frac{A}{B} = \lim_{y\to 0} - \frac{\sin(y)}{y} \cdot \frac{A}{B}. - -

- -

- Hence - \displaystyle\lim_{x \to 0}\frac{\sin(Ax)}{Bx}= - \dfrac{toFraction(A/B)[0]}{toFraction(A/B)[1]}. - -

- - -
- -
- -
-
- randRangeNonZero(0,9) -
- -

- Evaluate the following limit: -

- - \displaystyle\lim_{x \to A*A}\frac{x-A*A}{\sqrt{x}-A} - -

- -
- 2*A -
- -
-

Note we cannot evaluate this limit directly since the - denominator is zero when x=A*A.

- -

Multiply by the conjugate: -

- -

- \displaystyle\lim_{x \to A*A}\frac{x-A*A}{\sqrt{x}-A} = - \lim_{x \to A*A}\frac{x-A*A}{\sqrt{x}-A} \cdot\frac{\sqrt{x}+A}{\sqrt{x}+A} - -

- -

Now simplify. -

- \displaystyle\lim_{x \to A*A}\frac{x-A*A}{\sqrt{x}-A} = - \lim_{x \to A*A}\frac{(x-A*A)(\sqrt{x}+A)}{x-A*A} - -

- -

Since we are taking the limit as x\to A*A, we may assume that x\ne A*A. -

- -

Canceling factors we see: -

- \displaystyle\lim_{x \to A*A}\frac{x-A*A}{\sqrt{x}-A} = - \lim_{x \to A*A}(\sqrt{x}+A) - -

- -

We can evaluate this function at x = A*A. -

- -

Hence, we see the solution is 2*A -

- -
- -
-
-
-