Introduction to Mathematical Analysis I - Second Edition

122 4.4 L’HOSPITAL’S RULE Example 4.4.7 Consider the limit lim x → ∞ ln x x . Clearly the functions f ( x ) = ln x and g ( x ) = x satisfy the conditions of Theorem 4.4.5 . We have lim x → ∞ f 0 ( x ) g 0 ( x ) = lim x → ∞ 1 / x 1 = 0 It follows from Theorem 4.4.5 that lim x → ∞ ln x x = 0. Exercises 4.4.1 Use L’Hospital’s Rule to find the following limits (you may assume known all the relevant derivatives from calculus): (a) lim x →− 2 x 3 − 4 x 3 x 2 + 5 x − 2 . (b) lim x → 0 e x − e − x sin x cos x . (c) lim x → 1 x − 1 √ x + 1 − √ 2 . (d) lim x → 0 e x − e − x ln ( 1 + x ) . (e) lim x → 1 ln x sin ( π x ) . 4.4.2 For the problems below use L’Hospital’s rule as many times as appropriate to determine the limits. (a) lim x → 0 1 − cos2 x x sin x . (b) lim x → 0 ( x − π 2 ) 2 1 − sin x . (c) lim x → 0 x − arctan x x 3 . (d) lim x → 0 x − sin x x − tan x . 4.4.3 Use the relevant version of L’Hospital’s rule to compute each of the following limits. (a) lim x → ∞ 3 x 2 + 2 x + 7 4 x 2 − 6 x + 1 . (b) lim x → 0 + − ln x cot x .