100 Evaluate the following limits. Check your results by graphing. lim ln x x?? ?x

Solution Step 1 100 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: Suppose that we have one of the following cases, lim f(x)= or lim f(x)= ± xa g(x) 0 xa g(x) ± Where a can be any real number, infinity or negative infinity. In these cases we have lim f(x)= lim f(x) xa g(x) xa gx) Step 2 ln x00 Let us now evaluate lim x x By the direct substitution of x = we get indeterminate form. So we can apply l’hopital’s rule. 100 d(ln x0) lim ln x = lim dxd x x x dx(x) dx (ln x100) = 100(100x )9 x d 1 dx (x) = 2x d 100 1 99 ln x00 dxln x ) x100100x ) xm x = lix dx(x) = x 2 x 200x 200 200 = x x = x x = = 0 Thus lim ln x00= 0 x x