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MATH 131 Homework 2 (Due date: Thursday, Oct 17) Please do the following problems. Make sure to show all details of your work. 1. (16 pts) Find all real numbers x such that (4 pts for each part) (a) x 2 + 3x + 2 = 0. (b) x 2 + 4x + 2 = 0. 1 (c) x + = 2. x 1 (d) x + = 0. x 2. (10 pts) Let a and b be real numbers. Find all real numbers x (in terms of a and b) such that (x − a)(x − b) > 0, when (5 pts for each part) (a) a < b. (b) a = b. 3. (24 pts) Draw a graph of f and find lim x→2 f (x) where f : R → R is (6 pts each; 2 for the graph and 4 for the limit) (a) f (x) = 2x. (b) f (x) = |x − 2|. ¨ 4x − 4, when x ≤ 2, (c) f (x) = 4, when x > 2. ¨ x 2, when x 6= 2, p (d) f (x) = 2 2, when x = 2. 4. (12 pts) Use the rigorous definition of limit to justify your answer for parts (a) and (b) in the question above (6 pts for each part). 5. (4 pts) Let x, y, z be three real numbers, then the triangle inequality says that |x + y| ≤ |x| + | y|. (1) What does this statement says geometrically. In particular, why is it called the triangle inequality? 1 6. (extra credit; 8 pts) Prove the triangle inequality. (HINT. Divide into different cases and prove for each case separately. Alternatively, you can square both sides of (1). If you choose to do this, please justify why squaring both sides is allowed.) 7. (4 pts) Let f , g : R → R be functions and x 0 a real number such that the limits of f and g exist at x 0 . Use the triangle inequality to show that lim ( f (x) + g(x)) = lim f (x) + lim g(x). x→x 0 x→x 0 2 x→x 0