# 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