The Purplemath Forums

36: Find the limit: lim, x -> 0, (sin 4x)/(sin 6x)

I'm pretty sure I need to use the fact that lim, x->0, (sin x)/(x) = 1 (they gave us that in the section), but I don't see how...?

Thanks in advance

Thanks for the warning!

This is from my calculus book, but I think I only need algebra: Find the constant c that makes g continuous on (-infinity, +infinity): . / | x^2 - c^2 if x < 4 g(x) = < | cx + 20 if x >= 4 \ (The dot above doesn't mean anything, but the first line wouldn't line up without a leading character...

Prove that the function f(x) = x^101 + x^51 + x + 1 has neither a local maximum nor a local minimum.

I can take the derivative: f'(x) = 101x^100 + 51x^50 + 1

But what then?

The quadratic formula gives me x^50 = (-51+/-sqrt[(-51)^2-4(101)])/(2*101) = (-51+/-sqrt[2197])/(202) = (-51+/-46.9)/(202). The -51+/-46.9 will always be negative, so I can't do the 50th root of it to get a number for x. So this means I can't get a zero, so there isn't a critical point, so there can...

Use the Mean Value Theorem to prove the inequality |sin(a) - sin(b)| <= |a - b| for all a and b.

The MVT says that there is some c between a and b so sin(a) - sin(b) = cos(c)(a - b). Where do I go from there?

For what values of the numbers a and b does the function f(x)\, =\, axe^{bx^2} have the maximum value f(2)\, =\, 1 \mbox{?} f(2)\, =\, a(2)e^{b(2)^2}\, =\, 2ae^{4b}\, =\, 1 We've done the derivative tests. f'(x)\, =\, ae^{bx^2}\, +\, ax\left(2b...

Find the points on the ellipse 4x² + y² = 4 that are farthest away from the point (1, 0).

I must be forgetting something easy, to get started.

So I get a "distance" function of d(x)\, =\, 5\, -\, 2x\, -\, 3x^2 . The max/min points are the critical points, so d'(x)\, =\, -2\, -\, 6x\, =\, 0 , or x = -1/3. For x = 3, I get 4\left(-\frac{1}{3}\right)^2\, +\, y^2\, = 4 , so y\, =\, \pm \frac{4\sqrt{2}}{3} ...

The upper right-hand corner of a piece of paper, 12 inches long by 8 inches high, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? There is a right triangle folded down from the right-hand corner. The picture shows "x" as the amount folded...