MATH 210 Exam 2, Spring 2010

Exam 2-Math 210 1. (20 points) Use the ϵ -δ definition to show that lim x→3 x2 + 2x + 1 = 16. 1 2. (20 points) Let f : R → R be defined by f(x) = � x if x ∈ Q 0 if x /∈ Q . (a) Show that f(x) has a limit at x = 0 (b) Show that f does not have a limit at c for c ̸= 0. 2 3. (a) (20 points) Let f(x) = � 1 if x ∈ Q 0 if x /∈ Q . Show that f(x) is discontinuous everywhere in [0,1]. (b) Give a function f : [0, 1] → R that is discontinuous for every x ∈ [0, 1] while | f | is continuous for every x ∈ [0, 1]. 3 4. (20 points) Show that the equation x = cosx has a solution on the interval [0, π 2 ]. (Hint: Use the Intermediate Value Theorem). 4 5. (20 points) Show that f(x) = 1 x2 is uniformly continuous on [1, ∞) while it is not uniformly contin- uous on (0, ∞). 5