i have no clue how to start it. rationalizing it seems to tedious.

Question

how do i do this
thanks

evaluate the limit as h -> 0

((fourth root(3 + h) - 2(3+h) - fourth root(3) + 6))/h

Answer 1

Recall how you did your first exercises on derivatives of polynomials using the limit. Use the binomial theorem:

∜(3+h) = (3+h)^(1/4)
= 3^(1/4) + (1/4)(3^-3/4)(h) + (1/4)(-3/4)/2 (3^-7/4)(h^2) + ...

Now do your subtractions and collect terms:

The 3^(1/4) goes away
-2(3) + 6 goes away

and you end up with

(1/4 (3^-3/4) - 2)(h) + (1/4)(-3/4)/2 (3^-7/4)(h^2) + ...

Now divide that by h and you have

-2 + (1/4)(3^-3/4)(h) + ...

Now take the limit as h->0 and all the terms containing h vanish, leaving you with

-2 + (1/4)(3^-3/4)

Note that this is just the derivative of ∜x - 2x at x=3