# Choose the best answer for the given limit: `lim_(x->oo)` `(sqrt(x) -1)/(sqrt(3)x-1)` = A) -1 B) 0 C) `(2)/(3)` D) `(3)/(2)`

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### 2 Answers

The limit `lim_(x->oo) (sqrt x - 1)/(sqrt 3*x - 1)` has to be determined.

Substituting `x = oo` , gives a value of `oo` for the numerator as well as the denominator. The expression is `oo/oo` which is indeterminate. l'Hopital's rule can be used here to determine the limit. Substitute the numerator and the denominator by their derivatives.

This gives:

`lim_(x->oo) ((1/2)*(1/sqrt x))/sqrt 3`

As x tends to `oo` , 1/x tends to 0.

This gives the limit `lim_(x->oo) (sqrt x - 1)/(sqrt 3*x - 1) = 0`

We have given

`lim_(x->oo)(sqrt(x)-1)/(sqrt(3)x-1)`

Thus

`lim_(x->oo)(sqrt(x)(1-1/sqrt(x)))/(sqrt(x)(sqrt(3x)-1/sqrt(x)))`

`=1/oo`

`=0`

Thus

**B) 0 is most appropriate answer.**