HW2 Solutions

HW2 Solutions - ASSIGNMENT 2 SOLUTIONS Problem 1 a Both...

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ASSIGNMENT 2 - SOLUTIONS Problem 1. a) Both F 1 ( x ) and F 2 ( x ) have x = 0 as the possible vertical asymptote. We check by computing one-sided limits: lim x 0 + F 1 ( x ) = lim x 0 + x 2 + 1 + 1 x = , lim x 0 - F 1 ( x ) = lim x 0 - x 2 + 1 + 1 x = -∞ Either one of the above limits suffices to conclude that x = 0 is, indeed, a vertical asymptote for F 1 ( x ). But it’s a good idea to compute them both whenever possible, firstly because it’s good practice, and secondly because we’ll have to do that latter on when we’ll sketch graphs. Both lim x 0 + F 2 ( x ) and lim x 0 - F 2 ( x ) are of the form 0 0 . If we use the conjugate, we get: x 2 + 1 - 1 x = ( x 2 + 1 - 1)( x 2 + 1 + 1) x ( x 2 + 1 + 1) = x 2 x ( x 2 + 1 + 1) = x x 2 + 1 + 1 ( x = 0) We now see that lim x 0 + F 2 ( x ) = lim x 0 + x 2 + 1 - 1 x = lim x 0 + x x 2 + 1 + 1 = 0 and, similarly, lim x 0 - F 2 ( x ) = 0. Therefore x = 0 is not a vertical asymptote for F 2 ( x ). b) Comparing the degrees of the numerator and the denominator, we see that · α ( x ) has y = 0 as a horizontal asymptote; · β ( x

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