Math Question!?

Question

Without using graphing calculator, is it possible to find all the asymptotes of:

y = cos^2(x) / (x^2) ?

I can find one, when x = 0, but the book also has y=0. Any ideas how I can approach that?

Answer 1

Suppose f is a function. Then the line y = a is a horizontal asymptote for f if:

lim(x>>inf) f(x) = a or lim(x>> -inf) f(x) = a

Intuitively, this means that f(x) can be made as close as desired to a by making x big enough. How big is big enough depends on how close one wishes to make f(x) to a. This means that far out on the curve, the curve will be close to the line.

To find horizontal assymptotes you use limits.

Substitute in either x>> infinity or x>> negative infinity and if it equals to a constant then that is the assymptote.

Answer 2

cos^2 x is a bounded function

So 0<= cos^2 x <=1 forall x.

0< (cos^2 x)/x^2 <= 1/cos^2 x ........ forall x.

So, since 1/cos^2 x tends to 0 when x tends to infinity and so does 0 (the fact that it's 0 forall x values implies that its limit is in fact 0), then limit (cos^2 x)/x^2 = 0 too

Ilusion

Answer 3

Sure. Just let x get large without bound.

Doug