L'Hopitals Rule and Standard Limits

[Illusionist]

Homework Statement

Hello, I'm trying to find the limit(as n approaches 0) of [1-cos(n)]/(n^2). I have done a few of these before and haven't had to much trouble, but they all have been as n approaches infinity.

Homework Equations

I think the n approaches 0 is confusing me.

The Attempt at a Solution

I let n=0 and was left with 0/0, which would justify the use of L'Hopital's Rule. I then differentiated the numerator and denominator and was left with sin(n)/2n. Now do I let n=0, if so I would just be left with 0/0 again. I don't know where to go with this.

I am also having trouble with a another similar question, being finding the limit(as n approaches 0) of (n^2)sin^2(1/n). What sort of equations could I use as the less than and more than equations and how would the standard limits rules for n approaches infinity come into play?

Thanks in advance for any help or advice.

 

[danago]

Try applying L'Hopital's rule again, to Sin(n)/2n

 

[Illusionist]

Ok I see, thank you for that. Unfortunately I'm still having trouble with the second question.

 

[Illusionist]

I'm getting absolutely nowhere with the second question. I'm trying to find the limit of (n^2)sin^2(1/n).dx as n approaches 0 by using the Sandwich theorem.
IM a novice and inexperienced with the Sandwich thereom but I do understand it's principles. What sort of equations would be appropriate to use as the less than and greater than for the sandwich?
Any sort of help would be great and I'd appreciate it a bunch.

 

[Mindscrape]

What a minute, you got what happened with sin(n)/2n?

Hint:
Is (1-cos(n))/n^2 something that you can apply L'Hopital's rule to? What is cos(0)?

Sandwich thm is nice to know, but not needed here.

 

[Illusionist]

No I'm trying to do another question this time. I got the L'Hopital's Rule question but I'm now trying to solve the limit of (n^2)sin^2(1/n).dx as n approaches 0, by using the Sandwich Theorem.
I know for the Sandwich Theorem you must put a equation less than to the original on the left and greater than on the right, but I'm not sure what sort of equations to put in.
I tried putting 0 on the left, this would only work if the limit is zero I think, and 2(n^2)sin^2(1/n) on the right but don't know if this is right or where to go from here.
Sorry for the confusion.

 

[Mindscrape]

Oh, okay.

So the key to squeeze principle with trig functions is to start off with the basic trig inequalities such as
-1<= sin n <=1
and then add on to it

for example if you wanted to find the lim as n->infty of sin^2(5x)/(5-x) then you would start of with
-1<= sin5x <=1
0 <= sin^2(5x) <=1

then divide by the (5-x) and since x approaches infinity it will be negative and reverse the equalities

0/(5-x) >= sin^2(5x)/(5-x) >= 1/(5-x)

Then the limit as x->inft of 1/(5-x) = 0, so sin^2(5x)/(5-x) = 0.

Your problem is sort of similar, but has a multiplication instead of a division. Start off with
-1 <= sin(1/n) <= 1 (there is a discontinuity at n=0, but that is expected since we want to know that limit)
then build

 

[Illusionist]

Ok thanks for that. Let's see how I go:
-1<= sin n <=1
For my example I would start off with:
-1<= sin(1/n) <=1
0 <= sin^2(1/n) <=1
then multiply by n^2, gives:
0*(x^2) <= (x^2)sin^2(1/n) <=1*(n^2)

Now does this mean as n->inft of (n^2)=0?
Sorry I don't see how the right hand side limit would equal 0, did I go wrong somewhere?
Thanks again for that Mindscrape

 

[VietDao29]

...then multiply by n^2, gives:
0*(x^2) <= (x^2)sin^2(1/n) <=1*(n^2)

Now does this mean as n->inft of (n^2)=0?
Sorry I don't see how the right hand side limit would equal 0, did I go wrong somewhere?
Thanks again for that Mindscrape

The two last lines are wrong. You are throwing the x's out of nowhere. It should read:
0*(n²) <= (n²)sin^2(1/n) <=1*(n²)

Of course, the RHS limit is 0. Your n tends to 0, right? Not infinity. (It's what the problem stated, have a a closer look at what the problem says.) :)

 

[Mindscrape]

Maybe I should have made an example that even more closely resembled his problem, I think I might have confused him. I couldn't think of any good x->0 probs though.

Anyway, yes, with a few minor changes you got it.

 

[Gib Z]

I would directly calculate the taylor series from the taylor formula, then try work it out. Seems like it could help, but I am not sure.

 

[Illusionist]

Of course, the RHS limit is 0. Your n tends to 0, right? Not infinity. (It's what the problem stated, have a a closer look at what the problem says.) :)

Of course, my apologies. What a stupid mistake. Thank you so much everyone, very helpful.