Limit Calculation with L'Hospital's Rule: arctanx/arcsinx rigorously at x=0

In summary, the conversation discusses finding the limit of ##\frac{arctanx}{arcsinx}## as ##x## approaches 0, using L'Hospital's rule. The question is raised about whether knowing the derivatives of arcsin and arctan is necessary, and the idea of using Taylor series is suggested. The importance of knowing the derivatives for using l'Hospital's rule is emphasized.
  • #1
lep11
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##
 
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  • #2
Yes, since you solved it instantly with it.
 
  • #3
What if we 'don't know' the derivatives of arcsin and arctan?
 
  • #4
lep11 said:
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
Math_QED said:
Yes, since you solved it instantly with it.
I should check the conditions for l' Hospitals rule first.
Math_QED said:
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
lep11 said:
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.
 

FAQ: Limit Calculation with L'Hospital's Rule: arctanx/arcsinx rigorously at x=0

1. What is L'Hospital's Rule and how does it apply to limit calculations?

L'Hospital's Rule is a mathematical theorem that allows us to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of a function f(x) as x approaches a certain value is indeterminate, then the limit of the quotient of two functions f(x)/g(x) can be evaluated by taking the derivative of both functions and then evaluating the limit again.

2. What is the formula for applying L'Hospital's Rule to a limit calculation?

The formula for applying L'Hospital's Rule is:

lim [f(x)/g(x)] = lim [f'(x)/g'(x)]

where f'(x) and g'(x) are the derivatives of f(x) and g(x) respectively.

3. How can L'Hospital's Rule be applied to the limit of arctanx/arcsinx at x=0?

In this case, we can rewrite the limit as:

lim [arctanx/arcsinx] = lim [x/(1-cosx)]

Then, we can take the derivative of both the numerator and denominator:

lim [x/(1-cosx)] = lim [1/(sinx)]

Finally, we can evaluate the limit as x approaches 0, which gives us a value of 1.

4. Why is it important to use L'Hospital's Rule in this limit calculation?

Without L'Hospital's Rule, the limit of arctanx/arcsinx at x=0 would be undefined, as both the numerator and denominator approach 0. By applying L'Hospital's Rule, we are able to evaluate the limit and determine its value.

5. Are there any limitations to using L'Hospital's Rule in limit calculations?

Yes, L'Hospital's Rule can only be applied to indeterminate forms, such as 0/0 or ∞/∞. It cannot be used to evaluate limits of other types, such as infinite limits or limits at infinity. Additionally, it is important to check if the conditions for applying L'Hospital's Rule are satisfied, such as taking the limit as x approaches a specific value and ensuring that the functions are differentiable in the given interval.

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