4. Applications of Derivatives
Differentials
Problem 4.7.27
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0⁺ (1 - ln x) / (1 + ln x)
Verified step by step guidance1
Identify the limit to evaluate: lim_{x→0⁺} (1 - ln x) / (1 + ln x). As x approaches 0 from the right, ln x approaches -∞, leading to an indeterminate form of ∞/∞.
Since the limit results in an indeterminate form, apply l'Hôpital's Rule, which states that if lim_{x→c} f(x)/g(x) is of the form 0/0 or ∞/∞, then lim_{x→c} f(x)/g(x) = lim_{x→c} f'(x)/g'(x), provided the limit on the right exists.
Differentiate the numerator and denominator separately: f(x) = 1 - ln x and g(x) = 1 + ln x. Calculate f'(x) = -1/x and g'(x) = 1/x.
Substitute the derivatives back into the limit: lim_{x→0⁺} (-1/x) / (1/x) = lim_{x→0⁺} -1 = -1.
Conclude that the limit evaluates to -1, confirming that l'Hôpital's Rule was applied correctly and the limit exists.
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