x x (x ) Towers of exponents? The functions f(x) = (x ) and g(x) = x are different functions. For example, f(3) = 19.683and g(3) ? 7.6 × 10 . Determine whether lim+f(x)and lim g(x+ are indeterminate forms and evaluate the limits. x?0 x?0

Solution Step 1 x In this problem we are given with two functions namely f(x) = (x ) and g(x) = x x (x . We have to check whether lim f(x)and +im g(x)are in +ndeterminate forms and then x0 x0 we have to evaluate the limits. If they are in indeterminate form, we can evaluate the limit using l’hopital’s rule. l'Hôpital's Rule: f(x) 0 f(x) ± Suppose that we have one of the following cases, lim xa g(x) = 0r lixa g(x)= ± Where a can be any real number, infinity or negative infinity. f(x) f(x) In these cases we have lim g(x)= lim gx) xa xa Step 2 x Consider, f(x) = (x ) x Taking natural logarithm (ln)on both sides, we get x x ln f(x) = ln (x ) By using ln a = x ln a, we get x = x ln (x ) Again by using ln a = x ln a, we get 2 = x ln (x) ln (x) ln f(x) = 1 x Taking limit x 0 on both sides, we get ln (x) lim+ln f(x) = lim + 1 x0 x0 x2 By using lim(ln f(x)) = ln (lim f(x)), we get xa xa ln (lim f(x)) = lim ln (x) + + 2 x0 x0 x + Now substituting the limit x 0 in right hand side, we get ln (x) ln (lim f(x)) = lim 1 = which is an indeterminate form. x0 + x0 + x2 So we can apply l’hopital’s rule. ln (x) d (ln x) lim = lim dx + 2 + dx 2 x0 x x0 x d (ln x) = 1 dx x d ( 2 = 3 dx x x d ln (x) dxln x) Thus lim + 1 = lim + d( ) x0 x2 x0 dx x2 1 x = lim +2 x0 x3 x2 = lim + 2 x0 = 0 Therefore ln (lim f(x)) = 0 x0 + Raising to power e on both sides, we get 0 lim f(x) = e = 1 x0 + Hence lim f(x) = 1 x0 +