<p>\(\lim_{x \to 0} (1 + \sin x)^{\cot x} =\) ______</p>
Step-by-Step Solution
Key Concept: Recognize this as a 1^∞ indeterminate form and use the exponential limit technique: convert to e^(g(x)·ln(f(x))) form, then apply lim[u→0] ln(1+u)/u = 1.
<p><strong>Step 1:</strong> Identify the form. As x → 0: (1 + sin x) → 1 and cot x → ∞, giving 1^∞ (indeterminate).</p><p><strong>Step 2:</strong> Use the exponential form: Let y = (1 + sin x)^(cot x). Then ln y = cot x · ln(1 + sin x).</p><p><strong>Step 3:</strong> Evaluate the limit of the exponent:</p><p>lim[x→0] cot x · ln(1 + sin x) = lim[x→0] [ln(1 + sin x) / tan x]</p><p><strong>Step 4:</strong> Apply L'Hôpital's rule (0/0 form):</p><p>= lim[x→0] [cos x/(1 + sin x)] / sec²x</p><p>= lim[x→0] [cos x · cos²x] / (1 + sin x)</p><p>= (1 · 1) / (1 + 0) = 1</p><p><strong>Step 5:</strong> Therefore, lim[x→0] y = e^1 = <strong>e</strong></p>
Correct Answer: e