4. Applications of Derivatives
Differentials
Problem 61
Textbook Question
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0⁺ (cot x - 1/x)
Verified step by step guidance1
Identify the limit to evaluate: lim_{x→0⁺} (cot x - 1/x).
Rewrite cot x in terms of sine and cosine: cot x = cos x / sin x.
Combine the terms under a common denominator: cot x - 1/x = (cos x - sin x / x) / sin x.
Evaluate the limit as x approaches 0 from the right. Check if the limit results in an indeterminate form (0/0 or ∞/∞).
If an indeterminate form is found, apply l'Hôpital's Rule by differentiating the numerator and denominator separately, then re-evaluate the limit.
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