Need math help

[Ydoow]

Started Calc for summer classes and I\'m just confused/unsure of how this equation is written.

I\'m finding the limit as x approaches infinite of sin^2(x)/(x^2+1)

My only confusion is over sin^2(x). I\'ve never really encountered a trigonometric equation with the exponent in that position.

So, does sin^2(x) = sin(x) * sin(x) ? It doesn\'t look it to me. But I can\'t really test this in a calculator and online help just gives me ways to change it into cosines and sines. I can\'t do something that\'s not in this chapter or I\'ll lose points :l

This section is on horizontal asymptotes. So I believe I just have to find a ratio of which hits infinite faster, numerator or denominator.

Or if the limit doesn\'t even exist.

[RC1062]

Yep. Sin^2(x) = Sin(x) times Sin(x)

Its Sin^2(x) to denote the difference between that and Sin(x^2).

Hope that helps.

[Ydoow]

alright thanks. As it turns out, I don\'t know how to do this question lol.

Once I break it apart, not too sure what else to do.

Well, actually, I was thinking you could say -1<sin(x)<1, and then since sin is squared it would become 1<sin(x)<1

and then it would become 1/x^2+1 which would make it 1/infinite, making it equal 0

But...does that make sense? lol

[Wyrm]

If you\'re doing calculus, then you need to use the real meaning of limits.

lim_x?? f(x) = L iff (??>0)(?M>0) [x>M] ? [?>|f(x)-L|]

You want to show that lim_x?? (sin² x)/(x² + 1) = 0. This is tricky, because you cannot use the rule that if f(x) = sin² x converges to some L and g(x) = 1/(x² + 1) converges to K as x gets large, then f(x)g(x) converges to KL as x gets large, because f(x) = sin² x does not converge to anything as x gets large — it cycles between 1 and 0.

Here\'s a hint: show that, for any positive-definite function f(x), f(x) ? f(x) sin² x ? -f(x). That\'ll get you started.

[Exoscientist]

If you\'re doing calculus, then you need to use the real meaning of limits.

lim_x?? f(x) = L iff (??>0)(?M>0) [x>M] ? [?>|f(x)-L|]

You want to show that lim_x?? (sin² x)/(x² + 1) = 0. This is tricky, because you cannot use the rule that if f(x) = sin² x converges to some L and g(x) = 1/(x² + 1) converges to K as x gets large, then f(x)g(x) converges to KL as x gets large, because f(x) = sin² x does not converge to anything as x gets large — it cycles between 1 and 0.

Here\'s a hint: show that, for any positive-definite function f(x), f(x) ? f(x) sin² x ? -f(x). That\'ll get you started.

Hello, just joined the forum. How do you get the mathematical formulas into your posts?

Bob Clark

[Exoscientist]

Started Calc for summer classes and I\'m just confused/unsure of how this equation is written.

I\'m finding the limit as x approaches infinite of sin^2(x)/(x^2+1)

My only confusion is over sin^2(x). I\'ve never really encountered a trigonometric equation with the exponent in that position.

So, does sin^2(x) = sin(x) * sin(x) ? It doesn\'t look it to me. But I can\'t really test this in a calculator and online help just gives me ways to change it into cosines and sines. I can\'t do something that\'s not in this chapter or I\'ll lose points :l

This section is on horizontal asymptotes. So I believe I just have to find a ratio of which hits infinite faster, numerator or denominator.

Or if the limit doesn\'t even exist.

That\'s a good plan. That notation with the power as a superscript between the 'sin' and the '(x)' does indeed mean raise the sine function to that power. On a calculator you could enter like this: (sin(x))^2.

To get an idea of what the answer to your problem is try plugging large numbers on your calculator into: (sin(x))^2/(x^2+1)

Bob Clark

[Ydoow]

Ah got it. Went in for help, mainly because it was most convenient with my schedule and not because I didn\'t want to try it myself

I was close in a way. Except that sin^2(x) oscillates between 0 and 1.

So you end up with 0=<sin^2(x)<=1

then divide it all by (x^2+1) and 1/(x^2+1) approaches 0, meaning sin^2(x) is being 'crunched' between a number infinitely approaching 0, and 0.

So the limit is 0

Now that that\'s settled, time to prove the derivative of cot(x) = csc^2(x)

???

(I\'m asking in recitation in 10 minutes, so I shouldn\'t need help here)