Problem 1 solution: evaluate the following limits. Show your work. (a) lim x 0 (b) lim x + . 2x2 + 4x 1 x3 + 1. Solution: (a) the function f (x) = 2 cos x since f (x) is continuous at x = 0, we can evaluate the limit using substitution. X+1 2 is continuous for all x ( 1, 3) (3, + ). = 2 (b) this is a limit at in nity of a rational function. Our approach is to multiply the function by 1 x3 divided by itself and simplify: 2x2 + 4x 1 x3 + 1 lim x + . 2 x + 4 x3 + 1 x2 1. 1 xn = 0 for n > 0, we nd that: 2x2 + 4x 1 x3 + 1 lim x + x + 4. Problem 2 solution: compute the derivatives of the following functions and state where the derivative does not exist.
