Evaluate this limit...?

Question

lim x --> 1 2x^1/2 - x - 1/(x-1)^2

Answer 1

You need to learn to use brackets (parentheses) appropriately.

If you really mean lim(x->1) 2√ x - x - 1 / (x-1)^2, this is obviously -∞ since 2√ x - x --> 2√1 - 1 = 1 and 1/(x-1)^2 --> ∞.

However, I suspect you mean
lim(x-->1) (2√x - x - 1) / (x-1)^2, which is completely different.
lim(x-->1) (2√x - x - 1) / (x-1)^2
= lim(x-->1) -(x - 2√x + 1) / (x-1)^2
= lim(x-->1) -(√x - 1)^2 / [(√x - 1)(√x + 1)]^2
= lim(x-->1) -1 / (√x + 1)^2
= -1 / (1+1)^2 = -1/4.