First term test 8 mark triple TIPS question

[mrjoe2]

Homework Statement

(note: this is not an easy question and will be marked very strictly.)
Find the values of the constant a and b such that limx->0 ((ax+b)^1/3)-2)/x = 5/12
NOTE: you are not allowed to use L'Hospital's Rules for this question

Homework Equations

whatever you learned in calculus. there's not really specified equations to these questions

The Attempt at a Solution

finding b is quite simple. i know that 0/0 provides a hole in the graph of this function, so i want a b value that will give me such a hole because it will be the cause of the limit approaching that hole. that's my logic however skewed it may be, but i find it to be extraordinarily beneficial. so b would simplybe 8 since 8^1/3 is 2, and 2-2 is 0 and thus you get zero over 0. finding a is another story, i need some tips because i have no idea where to begin! thank you once again for saving my life lol..

 

[Dick]

This is again a difference quotient for the derivative of the function f(x)=(ax+8)^(1/3) evaluated at x=0. To prove it from first principles you have to go back and figure out how you proved (x^(1/3))'=(1/3)*x^(-2/3). Are you making up these silly rules that you can't use calculus to solve these limits? Because the whole point to calculus is to solve limits like this without having to go back and prove them from first principles every time.