"Limit"ed Edition

$\displaystyle \lim_{x \to\ 1} \dfrac {x}{x - 1} - \dfrac {1}{\ln x} = \infty - \infty$

Apply L'hopital's Rule:

$\displaystyle \lim_{x \to\ 1} \dfrac {x \ln x - x + 1}{(x - 1)(\ln x)} = \dfrac 00$

Apply L'hopital's Rule again:

$\displaystyle \lim_{x \to\ 1} \dfrac{1 + \ln x - 1}{\dfrac {x - 1}{x} + \ln x} = \dfrac 00$

And one final time:

$\displaystyle \lim_{x \to\ 1} \dfrac{\dfrac1x}{\dfrac{1}{x^2} + \dfrac 1x} = \dfrac 12$