limits problem please solve?

Question

If a real valued function g is defined by g (kx) =g(x) for all x and g (0) =1
And the real valued function h is defined by h (xy) =h(x) +h(y) for all x, y>0.
Find g(x) and h(x). And hence integrate f (xn+ x-n)*g(x)*h(x)/(1+x2) with respect
to x within the limits 0 and ∞, where f is any real valued function defined in the
limits of integration.

Answer 1

There are some parts missing.



For instance, is g supposed ot be continuous? What is k=?

Answer 2

First, g(k*1) = g(1) for all k, so let k = 0. This gives g(1) = g(0) and g(1) = g(k), so g(k) = 1 for all k. Hence g(x) = 1.



Next, h(x*1) = h(x) + h(1) so h(1) = 0. Since ln(x*y) = ln(x) + ln(y)we can set h(x) = ln(x)



The integral from 0 to infinity can be written as the sum of two integrals, the first form 0 to 1 and the second from 1 to infinity. On the first integral perform a change of variables: x -> 1/x and simplify. This gives the same integral as the second one, with a negative sign in front. Therefore the sum of these two integrals is zero.

Answer 3

The trick to limits of trig purposes is to precise them extra in terms of cos x, because of the fact it relatively is a million. So a sprint trig manipulation for the 1st one will supply us the respond: enable 3t = x, 6t=2x and the subject will become: tan(2x) / sin x Now, from the double perspective formula we've: cos 2x = cos² x - sin² x sin 2x = 2 sin x cos x so we get: tan 2x / sin x = [sin 2x / cos 2x] / sin x (2 sin x cos x) /(cos² x - sin² x) / sin x 2 cos x / (cos² x - sin² x) as t techniques 0 so does x, and so cos x techniques a million and sin x techniques 0 and we land up with: 2 (a million) /(a million-0) = 2 for the 2d: (sin² 4t) / (t²) = [ (sin 4t) / t ]² = [ (sin 2(2t)) / t ]² making use of the double perspective formula for perspective 2t we get: [ (sin 2(2t) ) / t ]² [ (2 sin 2t cos 2t) / t ]² making use of double perspective back we get: [ (2 (2 sin t cos t) (cos² t - sin² t)) / t ]² [ (4)(sin t /t) (cos t) (cos² t - sin² t)]² Now, making use of the nicely trouble-free cut back [(sin t ) / t ] = a million as t ? 0. (see the info on your calc e book) we get: [4(a million) (a million) (a million-0)]² =[(4)]² = sixteen