Solving Limits without L'Hospital's Rule

[sara_87]

Homework Statement

evaluate limit as x tends to 1:

(x^(2/3)-2x^(1/3)+1)/((x-1)^2)

Homework Equations

The Attempt at a Solution

=lim (x^(1/3)-1)^2/(x-1)^2

what do i do next??
(note, i don't want to use l'hospitals rule)

 

[wimma]

(note, i don't want to use l'hospitals rule)

...hmmm... why not?

Attached is my solution.

 

[Office_Shredder]

Substitute y = x^1/3 Then

$$\frac{(x^{1/3}-1)^2}{(x-1)^2} = \frac{(y-1)^2}{(y^3-1)^2}$$

and y³-1 = (y-1)(y² + y + 1)

 

[wimma]

Substitute y = x^1/3 Then

$$\frac{(x^{1/3}-1)^2}{(x-1)^2} = \frac{(y-1)^2}{(y^3-1)^2}$$

and y³-1 = (y-1)(y² + y + 1)

I can't believe that. I worked it out straight away but it took me like 30 mins to make that PDF lol... i wish i'd known this forum had LaTeX... then I might not have used LyX

Nike: Every day is a competition.

 

[sara_87]

Thank you very much wimma and office shredder.
:)

 

[wimma]

no problem XD