1. lim x->1 (x/x-1 - 1/ln x)

This is what I got so far...

=> lim x->1 (x-1)/(x-1 ln x)

=> lim x->1 d/dx (x-1)/d/dx (x-1 ln x)

=> lim x->1 1/x ln x .....I got stuck here

I check the book and the answer is 1/2 but how?

2. lim x->infinity ((squareroot x^2+x)-x)

This is what I got so far...

=> lim x->infinity d/dx (x^2+x^1/2 -x)

=> lim x->infinity d/dx (-x)/d/dx (x^2+x^-1/2) ...I got stuck here

The answer from the book is 1/2 but how?