Rationalizing the Numerator to Evaluate a Limit for a Tough Function

[emma3001]

Evaluate lim as x approaches a for (x^1/3 - a^1/3)/(x-a). I want to factor out x-a from the numerator so that the denominator is not equal to zero. How can I do this?

 

[Dick]

(A^3-B^3)=(A-B)(A^2+AB+B^2). Use that formula where A=x^(1/3) and B=a^(1/3). I.e. factor (x-a).

 

[HallsofIvy]

Another way to do that (Dick's hint is perfectly good) is not to factor but to multiply to rationalize the numerator. Using the same formula Dick gave, x- a= (x^1/3- a^1/3)(x^2/3+ a^1/3x^1/3+ a^2/3). Multiply both numerator and denominator by that.