Quick Calculus Question

[Stealth1024]

Can anyone find a way to solve this problem algebraicly?

Take the limit of the following as x approaches 1:

(x^(1/2) - x^2) / (1 - x^(1/2)


Even if you attempt to rationalize the numerator or denominator, the denominator still equals 0, so of course you can't solve this that way. The best thing I can come up with is to factor the denominator out of the numerator, however I'm not sure what the other factor would be. I am pretty sure the answer is 3.

 

[FeathersMcGraw]

Been a long time since I had calculus, but it looks like you're being asked to apply L Hopital's Rule.

 

[Stealth1024]

I don't think that will work either since if we take the derivative:


<P class=MsoNormal style="MARGIN: 0in 0in 0pt"><SPAN style="FONT-SIZE: 10pt; FONT-FAMILY: Arial">(x^(1/2) - x^2) / (1 - x^(1/2)<?xml:namespace prefix = o ns = "urn:schemas-microsoft-comfficeffice" /><o></o></SPAN>
<P class=MsoNormal style="MARGIN: 0in 0in 0pt"><SPAN style="FONT-SIZE: 10pt; FONT-FAMILY: Arial"><o> </o></SPAN>
<P class=MsoNormal style="MARGIN: 0in 0in 0pt"><SPAN style="FONT-SIZE: 10pt; FONT-FAMILY: Arial">(x^(-1/2) - x) / (x^-1) - x^(-1/2)

the denominator still comes out to 0... hmm</SPAN>

 

[Drakkon]

Its all about L'Hostpitals Rule (which was actually some other guys doing in the first place but i forget his name....possibly one of the bornoulli's?)
You would however get different aswers though depending on which side you approach i would think...I haven't graphed the function yet however

 

[FeathersMcGraw]

Originally posted by: Stealth1024

(x^(-1/2) - x) / (x^-1) - x^(-1/2)

the denominator still comes out to 0... hmm

I believe the error you're making is the following:

f'(x) where f(x) = 1 is not 1/x

 

[Stealth1024]

Elaborate Feathers... thanks

 

[FeathersMcGraw]

You're not properly applying the power rule.

d/dx (1) = d/dx (x^0) = 0 * (x^-1) = 0

The derivative of any constant value is zero.

 

[Whitecloak]

use L'Hospital's rule

if a function is of the form N/D where both N & D approach either infinity or zero at the same time when linits are taken, then N/D can be replaced by N'/D'

so

we

limit x-> 1 [(1/2)x^(-1/2)-2x]/[(-1/2)x^(-1/2)]

= (-3/2)/(-1/2)

= 3

 

[Stealth1024]

yep I realized its been an entire year before I've done derivation (I had integrated in the foreground of my mind) so once I did that correctly the principle worked great.

Thanks!