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Unformatted text preview: Test 1 - Solutions Math 102 Problem 1: ( 5 points ) Consider the function f ( x ) = 1 + 1 1 + 1 x . Compute the following limits: lim x f ( x ) , lim x - 1- f ( x ) , lim x f ( x ) . Solution: The easiest way to compute the limits is to first rewrite the function: f ( x ) = 1 + 1 1 + 1 x = 1 + 1 x + 1 x = 1 + x x + 1 = 2 x + 1 x + 1 Then lim x f ( x ) = lim x 2 x + 1 x + 1 = 1; lim x - 1- f ( x ) = lim x - 1- 2 x + 1 x + 1 = + ; lim x f ( x ) = lim x 2 x + 1 x + 1 = 2; 1 Problem 2: ( 5 points ) True or False? T F No rational function can have two horizontal asymptotes. T F No function can have more than two horizontal asymptotes. T F There exist differentiable functions which are not continuous. T F If f ( 3 ) = g ( 3 ) then the tangent lines to the graphs of f and g at x = 3 coincide. T F There exists a function which is continuous on [ , 1 ] and has a vertical asymptote at x = 1 2 . Solution: The answers are as follows: True: We know a useful rule telling us how to decide whether a rational function has horizontal asymptotes. According to that rule, there is either no asymptote (when the top has degree larger than the bottom), or one asymptote (when the top has same degree as the bottom, or when the top has...
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