lim t?a (t^(2/3) - a^(2/3))/(t-a)

Question

the original question is g(x)=t^(2/3) find g'(a) a≠0. I know how to find the derivative using the power rule, but I'm trying to help a Calc I student who hasn't started derivatives yet, and we're trying to figure out how to solve the limit.

Answer 1

To help make things a bit cleaner, I will define t^1/3 = x and a^1/3 = y. thus, the expression is

 

(x² - y²) / (x³ - y³)

 

Factor: the numerator is (x - y)(x + y), and the denominator is (x- y)(x² + xy + y²). This becomes

 

(x + y) / (x² + xy + y²) = (t^1/3 + a^1/3) / (t^2/3 + (ta)^1/3 + a^2/3)

 

In the limit t --> a, we have

 

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