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

  • #1
fishspawned
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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



CALCULUS%20CHAPTER%202%20TEST%2001%20Limits-8_zps15cgn22c.jpg
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Answers and Replies

  • #2
fresh_42
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Does the formula in the third attempt remind you on something?
 
  • #3
fishspawned
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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
fresh_42
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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
fishspawned
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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
fresh_42
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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
Delta2
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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
fishspawned
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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:
CALCULUS%20CHAPTER%202%20TEST%2001%20Limits-82_zpshpcpjng5.jpg
 
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  • #9
haruspex
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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:
CALCULUS%20CHAPTER%202%20TEST%2001%20Limits-82_zpshpcpjng5.jpg
Correct result.
 

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