# Trying to figure out a limit as x-->a

1. Jan 1, 2017

### fishspawned

1. The problem statement, all variables and given/known data

the limit x->a of [(x+2)^5/3 - (a+2)^5/3] / (x-a)

2. Relevant equations

limit laws

3. The attempt at a solution

2. Jan 1, 2017

### Staff: Mentor

Does the formula in the third attempt remind you on something?

3. Jan 2, 2017

### fishspawned

it looks like i'm trying to get a derivative. I am trying to do this without working using any derivatives and for some reason the power of 5/3 is completely messing with my head and i do not know how to proceed. i am still getting a 0/0 situation.

4. Jan 2, 2017

### Staff: Mentor

Well, differentiation seems to be the shortest way. Otherwise you probably will have to follow the paths the differentiating formulas are proven. The power $\frac{1}{3}$ is the difficulty here, for you cannot expand it easily. Why do you want to restrict yourself, once you already have the formula for the first derivative?

5. Jan 2, 2017

### fishspawned

The restriction is due to the question is apparently brought up before any derivatives are introduced. So apparently this is do-able without using straight up differentiation. Yes, the 1/3 is killing me. and i have no idea how to approach it.

6. Jan 2, 2017

### Staff: Mentor

I've just looked up how $\frac{d}{dx} x^{\alpha}=\alpha x^{\alpha-1}$ is proven for non-integer values $\alpha$.
It's done by the chain rule and $x^\alpha = \exp(\alpha \ln x)$. Perhaps this might help and you may use properties of the exponential function. Another substitution $y=x+2$ should decrease writing work.

7. Jan 2, 2017

### Delta²

Set $X=(x+2)^{1/3}$ $A=(a+2)^{1/3}$ then the it becomes $\lim_{X\rightarrow A}{\frac{X^5-A^5}{X^3-A^3}}$ . Hope I am right here and this helps.

8. Jan 2, 2017

### fishspawned

DELTA:
this seems to take care of it - provided you know how to expand a sum of powers to a high level - had to look it up to be honest:

so revised sheet:

9. Jan 7, 2017

### haruspex

Correct result.