Question

What is the limit, as x approaches 1, of (sqrt(x) - 1)/(x - 1)?

I need to show work, but I know the answer is 3/2, because I confirmed with a TI-89.

Answer 1

perhaps you have confused the answer with that of a different problem, the answer is not 3/2.

Since putting in x=1 gives 0/0, you can either use d'Hôpital's rule, or simply multiply by the conjugate (sqrt(x)+1) of the numerator to create the common factor (x-1).

So
(sqrt(x) - 1)/(x - 1)
=(sqrt(x)-1)(sqrt(x)+1)/[(x-1)(sqrt(x)+1)]
=(x-1)/[(x-1)(sqrt(x)+1)]
=1/(sqrt(x)+1)
=1/(sqrt(1)+1)
=1/2