math help thank u

Question

Find the limit if it exist. lim x→1^+ 1/lnx - 1/x-1

please be detailed thank you

Answer 1

lim_x→1+ [ 1 / lnx - 1 / (x-1) ] = lim_x→1+ [ (x - 1 - lnx) / ((x-1)lnx) ] Limit has the form 0/0.

= lim_x→1+ [ (1 - 1/x) / (lnx + (x-1)/x) ] (By L'Hopital's Rule)

Since the limit is still of the form 0/0, apply L'Hopital's Rule again to obtain:

lim_x→1+ [ (1/x²) / (1/x + 1/x²) ] = 1/2