Homework Help: Calculating a limit

1. Jun 12, 2010

gipc

I know I should apply L'Hopital's rule and use a^b=e^(b*ln(a)) but I can't finish the calculations.

limit as x->0 ((arcsin(x))/x) ^(1/x^2)

2. Jun 12, 2010

lanedance

thats a tricky one, so going with what you said
$$\lim_{x \to 0} (\frac{arcsin(x)}{x})^{\frac{1}{x^2}} = \lim_{x \to 0} e^{\frac{ln(\frac{arcsin(x)}{x})}{x^2}}$$

now let
$$b =\frac{ln(\frac{arcsin(x)}{x})}{x^2}$$

if the limit exists, its equal to e^(b), so finding the limit of a is sufficient

thats 0/0 indeterminate, so we can apply L'Hops rule - though i can see it will be a bit messy

Last edited: Jun 13, 2010
3. Jun 12, 2010

lanedance

4. Jun 12, 2010

gipc

sorry, still a no-go. can't get the algebra together. can someone please help? i've applied L'hopital's rule 3 times and it keeps getting uglier.

5. Jun 12, 2010

estro

I think it might be helpful considering Taylor Polynomials approximation.

6. Jun 12, 2010

gipc

We didn't learn yet the Taylor thingie. This assignment is about L'Hopital's rule.

7. Jun 13, 2010

lanedance

ok, so how about starting by looking at the arcsin function and its derivatives, lets abuse the notation a bit and call it a for brevity recognising its a function of x:
$$a(x) = arcsin(x), \ \ \ \ \ \ \lim_{x \to 0} a(x) = 0$$
$$a'(x) = (1-x^2)^{1/2}, \ \ \ \ \lim_{x \to 0} a'(x) = 1$$
$$a''(x) = x(1-x^2)^{3/2}, \ \ \ \lim_{x \to 0} a''(x) = 0$$
$$a'''(x) = (1-x^2)^{3/2} -3x(1-x^2)^{5/2}, \ \lim_{x \to 0} a'''(x) = 0$$

Last edited: Jun 13, 2010
8. Jun 13, 2010

lanedance

now going back to
$$b = \lim_{x \to 0}\frac{ln(\frac{arcsin(x)}{x})}{x^2} = \lim_{x \to 0} \frac{ln(a) - ln(x)}{x^2}$$

this is 0/0 so using L'Hop
$$= \lim_{x \to 0} \frac{a'/a - 1/x}{x^2} = \lim_{x \to 0}\frac{1}{2} \frac{a'x - a}{ax^2}$$

once again, this is 0/0 so using L'Hop
$$= \lim_{x \to 0}\frac{1}{2} \frac{a''x}{a'x^2+ 2ax}= \lim_{x \to 0}\frac{1}{2} \frac{a''}{a'x+ 2a}$$

one more time, this is 0/0 so using L'Hop
$$= \lim_{x \to 0}\frac{1}{2} \frac{a'''}{a''x+a'+ 2a'}= \lim_{x \to 0}\frac{1}{2} \frac{a'''}{a''x+3a'}$$

and at this point you should be able to sub in with the properties of the derivatives