Hello.

 

This is a question on my latest Calculus assignment:

 

Let f: R --> R be a function, and let a not equal zero. Suppose f is differentiable at a. Evaluate the following limit:

lim x-->a of (see attached image).

Express your answer in terms of f'(a).

 

Basically I'm having trouble understanding the third roots in the denominator, and how I am supposed to tackle this question. Any push in the right direction would be greatly appreciated.

The equation you have is very similar to the definition of a derivative at point a:

 

[math] f'(x)=Lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}[/math]

 

So if you can get rid of the cube roots you can just evaluate the derivative. To do this I would try and rationalize the denominator. This will give you:

 

[math]\frac{f(x)-f(a)}{x-a} \bullet (blah) [/math]

 

Factor the blah out and then evaluate the function.

 

I have not finished the solution yet, but this is the approach which I would take.