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sols1 - SOLUTIONS TO HOMEWORK ASSIGNMENT#1 1 Compute the...

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SOLUTIONS TO HOMEWORK ASSIGNMENT #1 1. Compute the following limits: (a) lim x →- 2 ( x 3 - 3 x 2 + 5) (b) lim x →- 1 (3 x 2 + 2 x + 1) 10 ( x 3 + 5) 5 (c) lim t 2 ( - 3 t 3 - 4 t + 5) 1 / 3 (d) lim t 3 t 2 - 9 t - 3 (e) lim t →- 3 t 3 - 9 t t + 3 (f) lim z 9 3 - z 9 - z SOLUTIONS: (a) lim x →- 2 ( x 3 - 3 x 2 + 5) = ( - 2) 3 - 3( - 2) 2 + 5 = - 15 by substituting x = - 2 and using continuity of the polynomial x 3 - 3 x 2 + 5 at x = - 2. (b) lim x →- 1 (3 x 2 + 2 x + 1) 10 ( x 3 + 5) 5 = (3( - 1) 2 + 2( - 1) + 1) 10 (( - 1) 3 + 5) 5 = 2 10 4 5 = 1 by substituting x = - 1 and using continuity of the function (3 x 2 + 2 x + 1) 10 ( x 3 + 5) 5 at x = - 1. (c) lim t 2 ( - 3 t 3 - 4 t + 5) 1 / 3 = ( - 3(2) 3 - 4(2) + 5) 1 / 3 = ( - 27) 1 / 3 = - 3 by substituting t = 2 and using continuity of the function ( - 3 t 3 - 4 t + 5) 1 / 3 at t = 2. (d) lim t 3 t 2 - 9 t - 3 = lim t 3 ( t - 3)( t + 3) t - 3 = lim t 3 ( t + 3) = 6. Notice that we can not put t = 3 right away as it leads to the nonsense 0 0 . This is why we must first do some algebra, namely cancel t - 3 from both denominator and numerator. Once this is done we can evaluate the limit by putting t = 3 and using continuity of the linear function t + 3.
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