# Math Help - Different kind of Limit Problem

1. ## Different kind of Limit Problem

Suppost that f is a differentiable function such that $f'(8)=2$. Without using L'Hospital's Rule, find the value of $\displaystyle\lim_{x\to 8}\frac{f(x)-f(8)}{x^{\frac{1}{3}}-2}$

2. Originally Posted by polymerase
Suppost that f is a differentiable function such that $f'(8)=2$. Without using L'Hospital's Rule, find the value of $\displaystyle\lim_{x\to 8}\frac{f(x)-f(8)}{x^{\frac{1}{3}}-2}$
does this help?

$f'(8) = 2 = \lim_{x \to 8} \frac {f(x) - f(8)}{x - 8} = \lim_{x \to 8} \frac {f(x) - f(8)}{\left( x^{\frac 13} - 2 \right) \left( x^{\frac 23} + 2x^{\frac 13} + 4 \right) }$

3. Originally Posted by Jhevon
does this help?

$f'(8) = 2 = \lim_{x \to 8} \frac {f(x) - f(8)}{x - 8} = \lim_{x \to 8} \frac {f(x) - f(8)}{\left( x^{\frac 13} - 2 \right) \left( x^{\frac 23} + 2x^{\frac 13} + 4 \right) }$
No not really, I still don't understand how to solve it. I get how you factored....thats about it

4. Originally Posted by polymerase
No not really, I still don't understand how to solve it. I get how you factored....thats about it
do you not see the limit we want in there?

$2 = \lim_{x \to 8} \frac {f(x) - f(8)}{\left( x^{\frac 13} - 2 \right) \left( x^{\frac 23} + 2x^{\frac 13} + 4 \right) } = \lim_{x \to 8} \frac {f(x) - f(8)}{x^{\frac 13} - 2} \cdot \frac 1{x^{\frac 23} + 2x^{\frac 13} + 4}$

5. Originally Posted by Jhevon
do you not see the limit we want in there?

$2 = \lim_{x \to 8} \frac {f(x) - f(8)}{\left( x^{\frac 13} - 2 \right) \left( x^{\frac 23} + 2x^{\frac 13} + 4 \right) } = \lim_{x \to 8} \frac {f(x) - f(8)}{x^{\frac 13} - 2} \cdot \frac 1{x^{\frac 23} + 2x^{\frac 13} + 4}$

Ahhh i get it...so the answer is 24 right? Ok back track a little...how did you know that 2 = the limit?

6. Originally Posted by polymerase
Ahhh i get it...so the answer is 24 right? Ok back track a little...how did you know that 2 = the limit?
yes, it's 24.

we were told f'(8) = 2. i used the limit definition of the derivative

7. Originally Posted by Jhevon
yes, it's 24.

we were told f'(8) = 2. i used the limit definition of the derivative
Ahhhhh....i get it completely now. Thanks