Solving limit using hopital rule

Question

lim as x tends to 0⁺ ( x^ln5/1+lnx) ) 

Answer 1

Let y = x^{[ln5/(1+lnx)]} , then lny = (ln5)(lnx)/(1+lnx)

 

lim as x tends to 0 of lny using L'Hopital's Rule becomes

 

lim as x tends to 0 of (ln5)(1/x)/(1/x) = ln5.

 

This is significant because lim as x tends to 0 of y = lim as x tends to 0 of e^lny = e^ln5 = 5.