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

[fishspawned]

Homework Statement

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

Homework Equations

limit laws

The Attempt at a Solution

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[fresh_42]

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

 

[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.

 

[fresh_42]

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?

 

[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.

 

[fresh_42]

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.

 

[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.

 

[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:

 

[haruspex]

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:

Correct result.