Why Is My Simple Limit Calculation Giving Different Results?

[PAR]

First of all, this is my first post on these forums, hello!

I need to to find this limit:

lim (x*sin(x))
x->0 (2(1-cos(x))

After applying l'Hopital's rule twice I get the answer 1.

However, when I ask Maple 11 to find the limit it returns 0 as the answer. I went to a bit farther and made Maple 11 apply l'Hopital's twice as well and it also came up with the answer being 1. In fact, after only one application of l'Hopital's rule, Maple returned 1 as the answer. My only conclusions are that either I don't fully understand the subtleties of l'Hopital's rule or that Maple is wrong.

First application of l'Hopital's rule gives:

lim (sin(x)+x*cos(x))
x->0 (2*sin(x))

Second application:

lim (2*cos(x)-x*sin(x))
x->0 (2*cos(x))

 

[laluser]

Okay i see why the answer is is 1. When you are applying L'Hôpital's rule make sure that you are getting the derivatives right. After you apply it twice and take the limit you should simplify and get 2/2 as your answer, hence the limit is equal to 1

 

[laluser]

oops. I thought Maple was returning the answer as 1. I don't really know why it is doing that. I'm sorry I tried...

 

[PAR]

Wow, so I guess Maple 11 has a bug in it, because when I tell it to find the limit of

(x*sin(x))
(2(1-cos(x))

as x -> 0 it tells me that it's 0. I guess this is a fair warning to those who are too reliant on computers to solve their math problems (like me). Thanks for the help!

edit: I found what the problem was, my first command when I had the problem was:

limit((sin(x)*(x))/(2(1-cos(x))),x=0);

I am missing a "*" between 2 and (1-cos(x)) this doesn't throw and error or anything, but it give the wrong answer. If I insert the "*" it gives the correct answer: 1. So I should have input:

limit((sin(x)*(x))/(2*(1-cos(x))),x=0);