How to find limit of sin^2(x)/x^2 as x--> infinity

[scorpa]

Hello Everyone,
I'm doing some practice problems and I am stuck on the following limits.
1) Limit as x approaches infinity of (sin^2(x))/x^2
Now for this one I thought that the answer would just be 1 seeing as the limit of sinx/x is 1, however the answer is apparently 0 and I can't see the reasoning for this.
2) Limit as x approaches 0 of (cosx-1)/xsinx
Now for this one I first tried getting an the identity for sin^2x out of it by squaring it but that didn't work. Then I just tried breaking it up, but I just keep ending up getting 0 but apparently the answer is 1/2.
3) Limit as x approaches pi/3 of (cosx-1/2)/(x-pi/3)
Again I am totally lost. I keep wanting to try and get an identity out of it but I just can't get anything useful.
4) Now this question I know should be easy. I am just having trouble understanding where to start. If F(x) = the integral of f(t)dt (the values on the integral are b=x and a=1) and f(t)= integral (squareroot(1+u^2))/u *du . Find F''(3).
Apparently the answer to this is (2squareroot82)/3
If this isn't clear I can always draw it out and post it, I'm sure that what I tryed to type isn't overly clear.
Sorry about these questions guys, I know I haven't exactly shown a lot of work, but these are the ones that I have no idea of how to do, I've been attempting them for 2 days now and just can't seem to get anywhere

 

[Dale]

1) Limit as x approaches infinity of (sin^2(x))/x^2
Now for this one I thought that the answer would just be 1 seeing as the limit of sinx/x is 1, however the answer is apparently 0 and I can't see the reasoning for this.

The limit of $$\frac{\sin (x)}{\Mfunction{x}}$$ is only 1 as x approaches 0, not as x approaches infinity. This problem is unrelated to that one.

The key to this one is to realize that $$\frac{{\sin (x)}^2}{x^2}$$is always between $$\frac{1}{x^2}$$ and 0.

I think you can use L'hopital's (sp?) rule for 2 and 3.

-Dale

 

[scorpa]

Hmmm ok, the way I interpreted that problem was in the form of (sinx)(sinx)/x^2 is this wrong? I always thought that sin^2x =(sinx)(sinx) and sinx^2/x^2 =$$\frac{{\sin (x)}^2}{x^2}$$

 

[Dale]

Sorry about the confusion:
$${\sin (x)}^2 = \sin (x)\,\sin (x) = {\sin }^2 (x)$$
Mathematica just uses the notation that I used while most textbooks favor your notation.

Either way, you are looking at the wrong part, in one case (sinx/x) you have the limit as x approaches 0, in the other (this problem) you have the limit as x approaches infinity. The squares have nothing to do with it in this case.

-Dale

 

[scorpa]

Oh ok I see, my mistake I see what you are getting at now. I am still having trouble with the other questions though, I just don't see where to start :(

 

[Dale]

I am still having trouble with the other questions though, I just don't see where to start :(

2 and 3 are just standard L'hopital's rule problems. Just apply his rule until you don't get 0/0.

I didn't really follow 4. I might be able to help if you could TEX it, but I might still be lost.

-Dale

 

[scorpa]

I will try and write it out and paste a link to number 4. As for 2 and 3 I don't actually know that rule so I will look it up on the net and see what I can come up with from that. Thanks

 

[scorpa]

Ok so with the L'Hopitals rule, all you have to do is take the derivative of the numerator and denominator and keep doing it until you don't get 0/0 anymore>? Does this always work like this? Can I use this rule all the time in a situation like this? Based on reading about the rule this is how I solved the questions and I ended up with the right answer:

2) cox-1/xsinx ---> -sinx/(sinx-xcosx)---> -cosx/cosx+cosx-xsinx which when you substitute in x=0 gives you an answer of -1/2!

3) (cosx-1/2)/(x-pi/3) -----> -sinx--->-sin(pi/3)---> -sqrt3/2!

If I can use that method all the time that is awesome! I never knew about it before, in class we were never shown that!

Thanks a lot!

 

[HallsofIvy]

If $$F(x)= \int_1^x f(t)dt$$ and [tex]f(t)= \int_1^{t^2} \frac{1+ u^2}{u}du[/itex],
find F'(3)[/tex]

Do you know the "Fundamental Theorem of Calculus": If $$F(x)= \int_1^x f(t)dt$$ the F'(x)=f(x). In particular, [tex]F'(3)=f(3)= \int_1^{9} \frac{1+ u^2}{u}du[/itex].

To do that integral, multiply both numerator and denominator by u to get
[tex]F'(3)= f(3)= \int_1^{9} \frac{u(1+ u^2)}{u^2}du[/itex], and let v= 1+ u^2.

 

[JasonRox]

Ok so with the L'Hopitals rule, all you have to do is take the derivative of the numerator and denominator and keep doing it until you don't get 0/0 anymore>? Does this always work like this? Can I use this rule all the time in a situation like this? Based on reading about the rule this is how I solved the questions and I ended up with the right answer:
2) cox-1/xsinx ---> -sinx/(sinx-xcosx)---> -cosx/cosx+cosx-xsinx which when you substitute in x=0 gives you an answer of -1/2!
3) (cosx-1/2)/(x-pi/3) -----> -sinx--->-sin(pi/3)---> -sqrt3/2!
If I can use that method all the time that is awesome! I never knew about it before, in class we were never shown that!
Thanks a lot!

See you were never shown that in class, I would recommend to only use it as a test. Your teacher probably expects you to find it through another route.

 

[scorpa]

Do you know the "Fundamental Theorem of Calculus": If $$F(x)= \int_1^x f(t)dt$$ the F'(x)=f(x). In particular, [tex]F'(3)=f(3)= \int_1^{9} \frac{1+ u^2}{u}du[/itex].
To do that integral, multiply both numerator and denominator by u to get
[tex]F'(3)= f(3)= \int_1^{9} \frac{u(1+ u^2)}{u^2}du[/itex], and let v= 1+ u^2.

Where does the square root go in that integral? And how do you get the second derivative from that? Stupid questions I know, but I'm just now seeing this one .


And how would you go about solving the limit problems without L'hopitals rule? Now that I learned it I rather like it...it's very handy!

 

[Dale]

Ok so with the L'Hopitals rule, all you have to do is take the derivative of the numerator and denominator and keep doing it until you don't get 0/0 anymore>? Does this always work like this? Can I use this rule all the time in a situation like this?

There are (of course) some limitations. If you are trying to take the limit of some f(x)/g(x) then the most important limitation is that the limit of f'(x)/g'(x) must exist.

For example, let $$f(x) = \sin (x)$$ and $$g(x) = x$$ then $$\Mfunction{Limit}(\frac{f(x)}{g(x)}, {x\rightarrow {\infty}}) = 0$$. However, $$f'(x) = \cos (x)$$ and $$g'(x) = 1$$ so $$\Mfunction{Limit}(\frac{f'(x)}{g'(x)}, {x\rightarrow {\infty}})$$ does not exist. So for this one L'Hopital's rule doesn't work. Instead you have to use the same method as in problem 1 to get the right answer.

-Dale

 

[VietDao29]

There are (of course) some limitations. If you are trying to take the limit of some f(x)/g(x) then the most important limitation is that the limit of f'(x)/g'(x) must exist.
For example, let $$f(x) = \sin (x)$$ and $$g(x) = x$$ then $$\Mfunction{Limit}(\frac{f(x)}{g(x)}, {x\rightarrow {\infty}}) = 0$$. However, $$f'(x) = \cos (x)$$ and $$g'(x) = 1$$ so $$\Mfunction{Limit}(\frac{f'(x)}{g'(x)}, {x\rightarrow {\infty}})$$ does not exist. So for this one L'Hopital's rule doesn't work. Instead you have to use the same method as in problem 1 to get the right answer.
-Dale

?
What is this?
What do you mean?
---------------------
L'Hopital rule is used when both numerator and denominator either tends to 0 or infinity, i.e:$$\frac{0}{0} \quad \mbox{or} \quad \frac{\infty}{\infty}$$. It states that, if:
$$\lim_{x \rightarrow \alpha} \frac{f(x)}{g(x)}$$ exists, and either:
$$\lim_{x \rightarrow \alpha} f(x) = 0 \quad \mbox{and} \quad \lim_{x \rightarrow \alpha} g(x) = 0$$ or:
$$\lim_{x \rightarrow \alpha} f(x) = \infty \quad \mbox{and} \quad \lim_{x \rightarrow \alpha} g(x) = \infty$$, then:
$$\lim_{x \rightarrow \alpha} \frac{f(x)}{g(x)} = \lim_{x \rightarrow \alpha} \frac{f'(x)}{g'(x)}$$.
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#1, you cannot use L'Hopital rule since you don't know what sin²(x) tends to when x tends to infinity. It can be done using Squeeze theorem, by noticing the fact that:
$$0 \leq \frac{\sin ^ 2 x}{x ^ 2} \leq \frac{1}{x ^ 2}$$
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#2, you have two different ways to do it without using L'Hopital rule:
The first way is to use power-reduction formulas:
$$\sin ^ 2 x = \frac{1 - \cos(2x)}{2}$$.
So here you have:
$$\lim_{x \rightarrow 0} \frac{\cos x - 1}{x \sin x} = \lim_{x \rightarrow 0} - \frac{2 \sin ^ 2 \left( \frac{x}{2} \right)}{x \sin x} = \lim_{x \rightarrow 0} - \frac{2 \sin ^ 2 \left( \frac{x}{2} \right)}{4 \frac{x}{2} \sin \left( \frac{x}{2} \right) \cos \left( \frac{x}{2} \right)}$$.
Can you go from here?
The second way is to multiply both numerator and denominator by (1 + cos x), so that you'll have sin²x in the numerator.
It goes like this:
$$\lim_{x \rightarrow 0} \frac{\cos x - 1}{x \sin x} = \lim_{x \rightarrow 0} \frac{(\cos x - 1)(1 + \cos x)}{x \sin x (1 + \cos x)} = \lim_{x \rightarrow 0} \frac{- \sin ^ 2 x}{x \sin x (1 + \cos x)}$$.
Can you go from here?
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#3, you can change 1 / 2 into: $$\frac{1}{2} = \cos \left( \frac{\pi}{3} \right)$$, then use the identity:
$$\cos \alpha - \cos \beta = -2 \sin \left( \frac{\alpha + \beta}{2} \right) \sin \left( \frac{\alpha - \beta}{2} \right)$$. Can you go from here?
L'Hopital rule is overkill... and as a matter of fact, you haven't learned it, so you shouldn't use it.
--------------------
If $$F(x) := \int_{\alpha} ^ {x} f(t) dt$$ then $$F'(x) = \frac{d \left( \int_{\alpha} ^ {x} f(t) dt \right)}{dx} = f(x)$$. There should be a proof in your book, or you can view it here (see part 3).
So $$F''(x) = f'(x) = \frac{d \left( \int_{1} ^ {x ^ 2} f(u) du \right)}{dx} \ = \ ?$$, where $$f(u) := \frac{\sqrt{1 + u ^ 2}}{u}$$. Note that the upper limit of the integral is x², not x.
Can you go from here?

 

[scorpa]

OK, thanks so much! I think I understand those limits now even without that handy little rule. And I can now get the answer for that integral! Thanks very much guys!