Calculation of limit

  • Thread starter lep11
  • Start date
  • #1
380
7

Homework Statement


Calculate ##\lim_{x \rightarrow 0} \frac{arctanx}{arcsinx}## 'rigoriously'.

The Attempt at a Solution


What's the best approach? L'Hospitals rule?

##\lim_{x \rightarrow 0} \frac{arctanx}{arcsinx}=\lim_{x \rightarrow 0} \frac{\sqrt{1-x^2}}{x^2+1} =1##
 

Answers and Replies

  • #2
member 587159
Yes, since you solved it instantly with it.
 
  • #3
380
7
What if we 'don't know' the derivatives of arcsin and arctan?
 
  • #4
member 587159
What if we 'don't know' the derivatives of arcsin and arctan?
Then you proof what these derivatives are equal too. It can't be solved in an easier way.
 
  • #5
380
7
Yes, since you solved it instantly with it.
I should check the conditions for l' Hospitals rule first.
Then you proof what these derivatives are equal too. It can't be solved in an easier way.
I am thinking whether I can assume we know the derivatives or begin with calculating the derivatives first?
It's kinda re-inventing the wheel though?

How about applying taylor series of arcsin and arctan?
 
  • #6
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,728
I should check the conditions for l' Hospitals rule first.

I am thinking whether I can assume we know the derivatives or begin with calculating the derivatives first?
It's kinda re-inventing the wheel though?

How about applying taylor series of arcsin and arctan?
How would you find the Taylor series without knowing the derivatives?

Nobody is re-inventing the wheel here. If you know the derivatives (or can find them easily) then l'Hospital's rule is useful; otherwise, it does you no good. In your case you know the derivatives, so l'Hospital works like a charm.
 

Related Threads on Calculation of limit

  • Last Post
Replies
2
Views
907
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
3
Views
941
Replies
3
Views
1K
  • Last Post
Replies
8
Views
868
Replies
3
Views
3K
Replies
3
Views
1K
Replies
2
Views
8K
Replies
4
Views
1K
Top