Okay, I attempted 3 of these and I don't know how to do the other two. So I hope I can get you guys to check if I am on the right track and teach me how to do the other two. Thanks a lot, a lot, a lot!
Use the airthmetic of limits, standard limits (clearly stated) or appropriate rules (clearly stated) to compute the limit of each sequence if it exists. Otherwise explain why the sequence diverges.
(a) =
I divided the whole thing with and using standard limits I got,
Then I used l'Hopital's rule to solve the limit for the and got 1/2 and just substituted it back in and my answer is 11/4.
(b)
I don't know how to do this one!
(c)
This one is to use sandwich rule right?
(d)
Used continuity rule and got,
divided the whole thing by n
= 0.5
(e)
I don't know how to do this one either.
For b)
Remember that the factorial function grows faster than polynomials or exponentials. The series diverges .
For c)
try this trick
Take the natural log of both sides
using log properties we get
We can now use L'hospitals rule to get
Now letting n go to infinity gives
For d you are correct
for e) rewrite as
and use L.H rule
Good luck.
Whoa, thanks for the many responses guys (: Took your guidance and attempted the questions!
For (c), this is how far I've got ..
Where do I go from here? Hmm.
As for (e), I followed TheEmptySet's advice and used L.H. rule, and this is what I've got,
After differentiating,
I don't know where to go from here too
You missed a minor point.
Try again... what is
LH rule can be a little dangerous here. Instead modify the question a little bit and see if you can recognize the limit...As for (e), I followed TheEmptySet's advice and used L.H. rule, and this is what I've got,
After differentiating,
I don't know where to go from here too
When , so lets call as .
Now where have I seen ?
If you havent seen this limit before, LH is still an option on this...
Well part of the problem is you shouldn't have ended up here.
Note that the orginial sequence was
Rewriting as
as this goes to
Now applying L'hospitials rule we get
This is where your error occured you forgot to use the chain rule when taking the derivative of the tangent function
Now when we reduce we get
I hope this clears it up.
Good luck.