MATH 2500: TAKE HOME 01 (30 points.) NAME: DUE: Wednesday, May 27th by 8 AM. DIRECTIONS: To receive full credit, make sure your work is neat and complete. 1. Determine the following limits algebraically. x2 1 x1 x 2 + x 1 (a) lim x2 1 x1 x 2 + x 2 (b) lim 3 1 6 x (c) lim x3 x 3 (d) lim t0 t +42 t (e) lim 2 csc2 (3) 0 10 2. Let f (x) = 1x . Circle the correct response or fill in the blank, as required. x 3 (a) For lim+ f (x), we are considering x-values that are a little bit x3 (b) The numerator of f (x), 1 x, approaches as x 3+ . (c) The denominator of f (x), x 3, becomes a very small (d) Hence, f (x) = 3. Let S() = greater than x = 3. less positive number as x 3+ . negative 1x positive becomes very 'large' (in absolute value) but so we write lim+ f (x) = negative x 3 x3 sin(2) . (a) Explain why S is not continuous at = 0. sin(2) . 0 (b) Find lim (c) Explain why the discontinuity at = 0 is removable, and remove it. S() = , if 6= 0 , if = 0 10 . 2x + k, if x 3 5. Sketch the graph of a function y = f (x) which satisfies all of the following criteria: lim f (x) = 3 x0 lim+ f (x) = lim f (x) = 3 x2 x0 The intervals of continuity of f are (, 0], (0, 2), (2, ) 6. Suppose x 2 + 1 f (x) cos(x) for all x in [1, 1]. Find lim f (x) and explain your reasoning. x0 10