1 ,0 ,? forms? Evaluate the following limits. Check your results by graphing. ln x lim ?x x? ?

Solution Step 1 In this problem we have to evaluate the limit lim ln x by using l'Hôpital's Rule when x x needed. l'Hôpital's Rule: f(x) 0 f(x) ± Suppose that we have one of the following cases, lim g(x)= o0 lim g(x)= ± xa xa Where a can be any real number, infinity or negative infinity. f(x) f(x) In these cases we have lim = lim xa g(x) xa gx) Step 2 Let us now evaluate lim ln x x x By the direct substitution of x = we get which is an indeterminate form. So we can apply l’hopital’s rule. 3 d(ln x) lim ln x = lim dxd x x x dxx) d (ln x) = 3ln x ( ) 1 dx x d 1 dx (x) = 2x d 3 2 1 ln x dx(ln x) 3 ln xx( ) Thus lix x = lx dx(x) = x 2 x 6xln (x) 6ln (x) = lim x = lim x = x x So we again apply l’hopital’s rule. 2 d 2 6ln (x) dx6ln (x)) xm x = x dxx) d 2 1 1 dx (6 ln x) = 6(2 ln x)( x = 12ln x( ) x d 1 dx (x) = 2x 6ln (x) dx6ln (x)) 12 ln xx( ) lim x = lim d(x) = lim 1 x x dx x 2x 24xln (x) 24 ln (x) = lim x = lim x x x = wich is an indeterminate form. So we again apply l’hopital’s rule. 24 ln (x) dx24 ln (x)) lim = lim d x x x dxx) d 24 dx (24 ln (x)) = x d 1 dx (x) = 2x d 24 lim 24 ln (x= lim dx24 ln (x)= lim x x x x dx x) x x 48x 48 1 = lx x = lx x= = 0 Thus lim ln x= 0 x x