Extention of l'hopital's rule?

[avr10]

Homework Statement

Hi all, I'm new here, and just have a quick question:

Evalute: lim as x$$\rightarrow$$$$\infty$$ of $$\frac{x - sin x}{x^{3}}$$

Homework Equations

l'hopital's rule?

The Attempt at a Solution

So far I've used L'hopital's rule to get it down to lim as x$$\rightarrow$$$$\infty$$ of $$\frac{cos x}{6}$$

but after that, I don't know what to do... I'm pretty sure that I have to use an extention of L'hopital's rule

thanks!

 

[evilpostingmong]

At infinite, cos(x)=1

 

[Dick]

At infinite, cos(x)=1

Don't be silly. cos(x) oscillates, it doesn't converge. And, arv10, if you are going to apply l'Hopital's rule multiple times, you have to make sure that at each stage you still have an infinity/infinity or 0/0 form. After the first application you have (1-cos(x))/(3x^2). That's not a form you can apply l'Hopital's rule to. You have to apply another technique to that. And once you've done that you should realize you didn't need to apply l'Hopital to begin with.

 

[buffordboy23]

At infinite, cos(x)=1

No. The squeeze theorem is your friend here.

 

[HallsofIvy]

Thought I had replied to this before.

You can't use L'Hopital's rule: The numerator does not go to either 0 or infinity: sin(x) oscilates between -1 and 1 and x goes to infinity.

But
$$\frac{x- sin x}{x^3}= \frac{x}{x^3}- \frac{sin x}{x^3}= \frac{1}{x^2}- \frac{sin x}{x^3}$$
The first fraction obviously goes to 0 and the second fraction, because sin x is always between -1 and 1, also goes to 0.

 

[Tobias Funke]

While I agree that L'Hospital's rule isn't necessary in this problem, it's not required that the fraction be of the form $$\infty/\infty$$. All that's required is that the denominator approaches infinity, so you can try L'Hospital as far as avr10 did, although it tells you nothing at the end since what you end up with has no limit.

 

[HallsofIvy]

Which, since this does have a limit, means that L'Hopital's rule does NOT apply here.

 

[Tobias Funke]

Well, it doesn't help you if you go all the way with it, but if for some reason you think the limit that Dick gets after using L'Hopital once is easier than the original, I would say it applies. Anyway, that's just semantics. I was just pointing out that the numerator not going to infinity doesn't rule out the limit from being a "candidate" to try L'Hopital.

 

[carlodelmundo]

Thought I had replied to this before.

You can't use L'Hopital's rule: The numerator does not go to either 0 or infinity: sin(x) oscilates between -1 and 1 and x goes to infinity.

But
$$\frac{x- sin x}{x^3}= \frac{x}{x^3}- \frac{sin x}{x^3}= \frac{1}{x^2}- \frac{sin x}{x^3}$$
The first fraction obviously goes to 0 and the second fraction, because sin x is always between -1 and 1, also goes to 0.

Correct me if I'm wrong (curious). But shouldn't the numerator "x - sin(x)" approach infinity? That is even if sin(x) oscilates from -1 to 1 ... the variable "x" is subtracted from "sin(x)" and as x increases... the subtraction from (-1 or 1) is so miniscule that x approaches infinity?

Graphically--y = x - sin(x) resembles a cubic function. Please shed some light.

Carlo

P.S. - I agree that the limit is 0.

 

[Dick]

Correct me if I'm wrong (curious). But shouldn't the numerator "x - sin(x)" approach infinity? That is even if sin(x) oscilates from -1 to 1 ... the variable "x" is subtracted from "sin(x)" and as x increases... the subtraction from (-1 or 1) is so miniscule that x approaches infinity?

Graphically--y = x - sin(x) resembles a cubic function. Please shed some light.

Carlo

P.S. - I agree that the limit is 0.

You CAN apply l'Hopital to (x-sin(x))/x^3. You get (1-cos(x))/(3x^2). You CAN'T apply l'Hopital to that.

 

[Tobias Funke]

You CAN apply l'Hopital to (x-sin(x))/x^3. You get (1-cos(x))/(3x^2). You CAN'T apply l'Hopital to that.

Sure you can. The denominator goes to infinity. You don't need the form $$\infty/\infty$$ to apply the rule (check baby Rudin). However, that's the last time that application of the rule yields something with a limit, so if you have an easier time showing that sinx/(6x) approaches 0, then you've successfully used L'Hospital to solve the problem, albeit not in the shortest way.