Find the limit of the sequence as n tends to 1

[sara_87]

Homework Statement

Find the limit of the sequence as n tends to 1

(3/(1-sqrt(x)) - (2/(1-cuberoot(x))

Homework Equations

The Attempt at a Solution

making a common denominator and expanding:
= lim $$\frac{1-3x^{1/3}+2x^{1/2}}{1-x^{1/3}-x^{1/2}+x^{1/6}}$$

I then divided the whole thing by x^1/2 but didnt get anywhere.
Any help would be v much appreciated.
thank you

 

[gabbagabbahey]

= lim $$\frac{1-3x^{1/3}+2x^{1/2}}{1-x^{1/3}-x^{1/2}+x^{1/6}}$$

Simply substituting x=1 into this gives 0/0, so your limit is in one of the forms that qualify for l'hopital's rule...

 

[sara_87]

ok,thanks so i used L'hospital's rule and got that the limit is 0; is that right?

 

[gabbagabbahey]

Nope, you shouldn't be getting zero...there is actually an error in your first post: $$x^{1/3}x^{1/2}=x^{1/3+1/2}=x^{5/6} \neq x^{1/6}$$

You will have to use L'hopistal's rule twice

 

[sara_87]

you're right
but in that case, when i use l'hospitals rule once, i get 0/-0.5 = 0
so why must i use l'hospital's rule again?

 

[gabbagabbahey]

You should be getting 0/0 after the first time (5/6-1/3-1/2=0 not -0.5)

 

[sara_87]

Oh my, I am so stupid! i should have known that.
Yep, i see my mistake.
L'hospital's rule twice gives limit =1/2
right?

 

[gabbagabbahey]

Yup!

 

[sara_87]

Thanks.