APPM 1350 Final: appm1350fall2014examfinal_sol
Document Summary
Fall 2014: the following parts are not related: (a)(5 pts) evaluate the limit lim x (b)(5 pts) evaluate the limit lim x 0+ cosh(x)1/x. Justify your answer. x4 cos(cid:18) 2 (c)(7 pts) prove that lim x 0 (d)(8 pts) suppose f (x) = x2, x 2. Use the limit de nition of the derivative to determine whether or not f (x) is di erentiable at the point x = 2. You may not use l"hospital"s rule for this problem. Solution: (a)(5 pts) we have the indeterminate form 0 and so if let y = lim using continuity yields x cosh(x)1/x then taking the natural log and ln(y) = ln(cid:16) lim x cosh(x)1/x(cid:17) = lim x . = lim x tanh(x) = 1 x and so lim x cosh(x)1/x = y = e1 = e. (b)(5 pts) here we have the indeterminate from + . Combining the natural log terms and using continuity yields lim x 0+
