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19. lim x } 1 l / n ˇ x x w } 5 lim x } ˇ x w 1 ln x } 5 0 } ˇ 1 x w } grows slower than ln x as x . 20. lim x } l e n 2 x x } 5 lim x } e x 1 ln x } 5 0 e 2 x grows slower than ln x as x . 21. lim x } x 2 ln 2 x ln x }5 lim x 1 } ln x x } 2 2 2 5 lim x 1 } 1 1 / x } 2 2 2 5 lim x ( x 2 2) 5‘ x 2 2 ln x grows faster than ln x as x . 22. lim x } 5 ln ln x x } 5 5 5 ln x grows at the same rate as ln x as x . 23. Compare e x to x x . lim x } e x x x } 5 lim x 1 } e x } 2 x 5 0 e x grows slower than x x . Compare e x to (ln x ) x . lim x } (ln e x x ) x } 5 lim x 1 } ln e x } 2 x 5 0 e x grows slower than (ln x ) x . Compare e x to e x /2 . lim x } e e x x /2 } 5 lim x e x /2 e x grows faster than e x /2 . Compare x x to (ln x ) x . lim x } (ln x x x ) x } 5 lim x 1 } ln x x } 2 x since lim x } ln x x } 5 lim x } 1 1 / x }5‘ . x x grows faster than (ln x ) x . Thus, in order from slowest-growing to fastest-growing, we get e x /2 , e x , (ln x ) x , x x . 24. Compare 2 x to x 2 . lim x } 2 x 2 x } 5 lim x } (ln 2 2 x )2 x } 5 lim x } (ln 2 2 ) 2 2 x 2 x grows faster than x 2 . Compare 2 x to (ln 2) x . lim x } (ln 2 x 2) x } 5 lim x 1 } ln 2 2 } 2 x since } ln 2 2 } . 1. 2 x grows faster than (ln 2) x . Compare 2 x to e x . lim x } 2 e x x } 5 lim x 1 } 2 e } 2 x 5 0 since } 2 e } , 1. 2 x grows slower than e x . Compare x 2 to (ln 2) x . lim x } (ln x 2 2) x since lim x x 2 and lim x (ln 2) x 5 0. x 2 grows faster than (ln 2) x . Thus, in order from slowest-growing to fastest-growing, we get (ln 2) x , x 2 ,2 x , e x . 25. Compare f 1 to f 2 . lim x } f f 2 1 ( ( x x ) ) } 5 lim x } ˇ 1 w ˇ 0 w x x w w 1 w 1 w lim x ! 1 § 0 § 1 § } 1 x § } § 5 ˇ 1 w 0 w Thus f 1 and f 2 grow at the same rate. Compare f 1 to f 3 . lim x } f f 3 1 ( ( x x ) ) } 5 lim x } ˇ ˇ x w 1 w x w 1 w } 5 lim x ! 1 § 1 § } 1 x § } § 5 1 Thus f 1 and f 3 grow at the same rate. By transitivity, f 2 and f 3 grow at the same rate, so all three functions grow at the same rate as x . 26. Compare f 1 to f 2 . lim x } f f 2 1 ( ( x x ) ) } 5 lim x } ˇ x x 4 w 2 1 w x w } 5 lim x ! 1 § 1 § } x 1 § 3 } § 5 1 Thus f 1 and f 2 grow at the same rate. Compare f 1 to f 3 . lim x } f f 3 1 ( ( x x ) ) } 5 lim x } ˇ x 4 w x 2 2 w x w 3 w lim x ! 1 § 2 § } 1 x § } § 5 1 Thus f 1 and f 3 grow at the same rate. By transitivity, f 2 and f 3 grow at the same rate, so all three functions grow at the same rate as x . 27. Compare f 1 to f 2 . lim x } f f 2 1 ( ( x x ) ) } 5 lim x } ˇ 9 w x 3 w x 1 w 2 w x w } 5 lim x } ˇ 9 w ˇ x w 9 w 1 w x w 2 w x w } 5 lim x ! 1 § 1 § 1 § } 2 9 } § 2 x § 5 1 Thus f 1 and f 2 grow at the same rate. Compare f 1 to f 3 . lim x } f f 3 1 ( ( x x ) ) } 5 lim x } ˇ 9 w x 3 w 2 w x 4 w x w } 5 lim x } ˇ 9 w ˇ x w 9 w 2 w x w 4 w x w } 5 lim x !

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