Pre-Calc Homework Solutions 333

Pre-Calc Homework Solutions 333 - Section 8.2 36. (a) Apply...

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36. (a) Apply L’Hôpital’s Rule n times to find lim x . lim x 5 lim x } a e n n x ! }5‘ Thus e x grows faster than a n x n 1 a n 2 1 x n 2 1 1 1 a 1 x 1 a 0 as x . (b) Apply L’Hôpital’s Rule n times to find lim x . lim x 5 5 lim x } (ln a n a n ) n ! a x }5‘ Thus a x grows faster than a n x n 1 a n 2 1 x n 2 1 1 1 a 1 x 1 a 0 as x . 37. (a) lim x } l x n 1/ x n } 5 lim x 5 lim x } x 1 n / n } 5 0 Thus ln x grows slower than x 1/ n as x for any positive integer n . (b) lim x } ln x a x } 5 lim x 5 lim x } a 1 x a } 5 0 Thus ln x grows slower than x a as x for any number a . 0. 38. lim x 5 lim x 5 lim x 5 0 Thus ln x grows slower than any nonconstant polynomial as x . 39. Compare n log 2 n to n 3/2 as n . lim n } n l n o 3 g /2 2 n } 5 lim n } lo n g 1 2 /2 n } 5 lim n 5 lim n 5
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