Find the limit of (sqrt(1+x) - 1)/(cuberoot(1+x)-1) as x->0

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Find the limit of (sqrt(1+x) - 1)/(cuberoot(1+x)-1) as x-->0

Homework Statement



Fin limit of the follwing as x tends to 0:

[sqrt(1+x)-1]/[cuberoot(1+x)-1]

Homework Equations





The Attempt at a Solution



using change of variables:
let y=sqrt(1+x)
so = limit (as y tends to 1): [y-1]/cuberoot(y^2)-1]

Then i got stuck.

any help woyld be v much appreciated.
Thank you.
 

Answers and Replies

  • #2
malawi_glenn
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have you done taylor series?
 
  • #3
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Yes but we have to do this by changing the variables.
 
  • #4
HallsofIvy
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Looks to me like a problem where you would want to "rationalize the denominator.
[itex]a^3- b^3= (a- b)(a^2+ ab+ b^2)[/itex] so to "rationalize" [itex]^3\sqrt{y^2}- 1[/itex], you need to multiply by [itex]^3\sqrt{y^4}+ ^3\sqrt{y^2}+ 1[/itex]. Multiplying the numerator and denominator by that will make the denominator [itex]y^2- 1[/itex]. You should be able to factor a "y- 1" out of the numerator also to cancel the "y- 1" in the denominator,.
 
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  • #5
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Ok so after i cancel the y-1, i get:
(cuberoot(y^2)+1)/(y+1)
so substituting the y=1, i should get limit =1
is that right?
 
  • #6
HallsofIvy
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Assuming you the calculations correctly, which you did not show, yes.
 
  • #7
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since after cancelling the y-1,this will give: (cuberoot(y^2)+1)/(y+1)=2/2=1

Thank you.
 

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