Limit of a function as x goes to 1

  • Thread starter nothingkwt
  • Start date
  • #1
33
0

Main Question or Discussion Point

f(x) = 1-cos(x-1) / (x-1)2

Can someone please explain how this can be done other than L'Hospital's rule? I tried but I just don't understand how. I know the answer is 1/2 though.

I tried separating to 2 limits but i got ∞ - ∞ and that is obviously wrong but I don't know what my mistake was.
 

Answers and Replies

  • #2
418
28
You could do this by doing a series expansion for the numerator and looking at the limit of the leading order expression. This is a useful series expansion:

cos(x-1) ≈ 1 - (x-1)2/2! + (x-1)4/4! + ...
This is basically the definition of cosine, but you can get it through taylor expansion about x=1. Note that as x goes to 1, the higher order terms are negligible.
If you plug this into the numerator of your expression, you get some nice cancellation happening.
 
  • #3
34,071
9,962
There are brackets missing for the numerator, I think.

If you know good upper and lower estimates for cos() around 0, you can use them. The full taylor series is the better solution, if you can use that.
 
  • #4
33,310
5,003
A Taylor's Series expansion seems to me to be the way to go.
 
  • #5
559
8
A Taylor's Series expansion seems to me to be the way to go.
How would you go about even expressing that as a taylor series o_O
 
  • #6
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
3,750
99
iRaid, can you express cos(x-1) as a Taylor series around x=1?

Actually Jolb has it above, but it doesn't do much if you don't know why that is what it is.
 

Related Threads on Limit of a function as x goes to 1

  • Last Post
Replies
2
Views
1K
Replies
4
Views
8K
Replies
5
Views
1K
Replies
1
Views
2K
Replies
3
Views
1K
Replies
4
Views
2K
Replies
6
Views
943
Top