Find the limit. Use l'Hospital's Rule where appropriate.?

Question

If there is a more elementary method, consider using that. If l'Hospital's Rule doesn't apply, explain why.

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

Answer 1

First, perform the subtraction to obtain:
(x - 1 - ln x)/[(x - 1)ln x]

And take it from there, for we have indeterminate form 0/0.

Answer 2

= lim x=>1 (x-1-lnx)/(x-1)lnx.
as you should know lnx/(x-1)=>1
so we an subtitute lnx/Where it is a factor ) by (x-1
so we get
lim(x-1-lnx)/(x-1)^2 and now we must go to L´Hôpital
= lim( 1-1/x)/2(x-1) = lim 1/2x= 1/2