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Midterm1 Solutions - Test 1 Solutions Math 102 Problem 1(5...

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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 0 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 0 f ( x ) = lim x 0 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
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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 0 ( 3 ) = g 0 ( 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 [ 0 , 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
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