Limit of (x^3-1)/(x^1/2-1) as x approaches 1

[tg43fly]

Homework Statement

lim x->1 (x^3-1)/(x^1/2-1)
ans:6

Homework Equations

The Attempt at a Solution

(x^1/2-1)^-1
can be converted into (plugged into wolfram):
(-x^1/2-1)/(1-x)
i want to how this done if I'm factorize out the 0 in the denominator

 

[lendav_rott]

If I'm getting it right you have

Numerator X³ -1
Denominator Sqrt(x) - 1

How can you factorize X³-1? If you know how the answer stares you right in the face.

 

[tg43fly]

ok i may have expressed myself wrongly there.
i wanted to know is how (1 / sqrt(x) - 1) can be converted into (-sqrt(x) - 1) / (1 - x)
noticed it was just multiplying denominator and numerator by its conjugate.
so yeah, should've noticed the elephant in the room.
thanks for the help anyway.

 

[lendav_rott]

same way as 1/sqrt2 is sqrt2/2

you lose square root in the denominator
1*(sqrt(x) +1)/ (sqrt(x) -1)(sqrt(x) +1)
sqrt(x) +1 / (x-1) , multiply both sides of the division sign by -1 and you arrive at what you are looking for.

 

[Ray Vickson]

Homework Statement

lim x->1 (x^3-1)/(x^1/2-1)
ans:6

Homework Equations

The Attempt at a Solution

(x^1/2-1)^-1
can be converted into (plugged into wolfram):
(-x^1/2-1)/(1-x)
i want to how this done if I'm factorize out the 0 in the denominator

Why not let t = x^(1/2), and so have the limit of (t^6 - 1)/(t-1) as t → 1?