Introduction to Mathematical Analysis I - Second Edition

121 It now follows from Theorem 4.4.2 that lim x → 0 ln x 2 1 + 1 3 √ x 2 = 0 . Remark 4.4.3 The proofs of Theorem 4.4.1 and Theorem 4.4.2 show that the results in these theorems can be applied for left-hand and right-hand limits. Moreover, the results can also be modified to include the case when ¯ x is an endpoint of the domain of the functions f and g . The following theorem can be proved following the method in the proof of Theorem 4.4.1 . Theorem 4.4.4 Let f and g be differentiable on ( a , ∞ ) . Suppose g 0 ( x ) 6 = 0 for all x ∈ ( a , ∞ ) and lim x → ∞ f ( x ) = lim x → ∞ g ( x ) = 0 . If ` ∈ R and lim x → ∞ f 0 ( x ) g 0 ( x ) = `, then lim x → ∞ f ( x ) g ( x ) = `. Example 4.4.6 Consider the limit lim x → ∞ 1 x ( π 2 − arctan x ) . Writing the quotient in the form 1 / x π 2 − arctan x we can apply Theorem 4.4.4 . We now compute the limit of the quotient of the derivatives lim x → ∞ − 1 / x 2 − 1 x 2 + 1 = lim x → ∞ x 2 + 1 x 2 = 1 . In view of Theorem 4.4.4 the desired limit is also 1. The following theorem can be proved following the method in the proof of Theorem 4.4.2 . Theorem 4.4.5 Let f and g be differentiable on ( a , ∞ ) . Suppose g 0 ( x ) 6 = 0 for all x ∈ ( a , ∞ ) and lim x → ∞ f ( x ) = lim x → ∞ g ( x ) = ∞ . If ` ∈ R and lim x → ∞ f 0 ( x ) g 0 ( x ) = `, then lim x → ∞ f ( x ) g ( x ) = `.