# Find limit as x-> a of this function

mathwiz123
This is one of the actual example L'hopital used in his book "L'analyse des Infiniment Petits Pour I'Intelligence des Lignes Courbes"

Check it out!

Find the limit of x as it approaches a for

y=[[(2*a^3*x-x^4)^.5]-[a^3(a^2x)^.5]]/(a-(ax^3)^.25)

Unfortunately, I can't get rid of the indeterminate form! Ideas/help?

## Answers and Replies

Saketh
I just want to make sure - is this the problem?
$$y = \frac{\left ( \sqrt{2a^3x-x^4}-a^3\sqrt{a^{2x}} \right )}{a - \sqrt{ax^3}}$$

mathwiz123
l'hopital

The diviser should be [a-(ax^3)^.25]

stunner5000pt
$$y = \frac{\left ( \sqrt{2a^3x-x^4}-a^3\sqrt{a^{2x}} \right )}{a - \sqrt{\sqrt{ax^3}}}$$

mathwiz123
and instead of a^3(a*a*x), it should be a(a*a*x)^(1/3)

stunner5000pt
i wonder if flat out differentiaon would just do it...

maybe you rationalize the numerator?

mathwiz123
I think a little simplification after rationalization should do...hmmm. By the way, how do you do that math font?

$$y = \frac{\left ( \sqrt{2a^3x-x^4}-a\sqrt{a^{2x}} \right )}{a - \sqrt{ax^3}}$$

mathwiz123
Thanks! And just for clarification. This is what I was trying to find.
$$y = \frac{\left ( \sqrt{2a^3x-x^4}-a\sqrt{a^{2}x} \right )}{a - \sqrt{ax^3}}[\tex] courtrigrad [tex] y = \frac{\left ( \sqrt{2a^3x-x^4}-a\sqrt{a^{2}x} \right )}{a - \sqrt{ax^3}}$$

mathwiz123
Yes! exactly. Thanks for the help.

$$a^4 -ax^3$$