Evaluate: lim ((x^(1/3)-a^(1/3))/(x-a)) as x->a

Question

How to get 1/3a^(2/3)

Answer 1

(x^1/3-a^1/3) / (x-a)

 

Let's do a substitution to make the math a little easier.

 

Let y = x^1/3 and let b = a^1/3

 

So this equals (y - b) / (y³ - b³)

 

Factor the denominator:

(y - b) / ((y - b)(y² + by + b²)

 

Divide through by (y - b):

1 / (y² + by + b²)

 

Substitute back in:

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

 

As x approaches a, this approaches:

1 / (a^2/3 + a^2/3 + a^2/3) =

1 / (3a^2/3)

 

...which is what you were trying to get to.