solve the limit of x approaching 0+ of lnx+1/ln(x+1)

Question

this is how I solved it but I dont know if it's correct https://ibb.co/5vLQ0C5

this is the limit to solve https://ibb.co/5vLQ0C5

Answer 1

rewrite ln(x)+1/ln(x+1)=ln(x)[1+((1/ln(x+1))/ln(x)]

now L=lim_x->0+ ln(x)[1+((1/ln(x+1))/ln(x)]

which is L=(-infinity)[(-infinity)/0⁺]=(-infinity)(-infinity)=+infinity