Limits, Continuity & Differentiability
Indeterminate Forms
Grade 12
Question:
<p>The value of <span>\(\lim_{x \to 0}(\sin x)^x\)</span> is</p>
<p>(a) <span>\(1\)</span></p>
<p>(b) <span>\(e\)</span></p>
<p>(c) <span>\(e^{-1}\)</span></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use logarithm to convert the indeterminate form and apply L'Hôpital's rule or Taylor series
<p>As <span>\(x \to 0\)</span>, <span>\((\sin x)^x = e^{x \ln(\sin x)}\)</span>. We need <span>\(\lim_{x \to 0} x \ln(\sin x) = \lim_{x \to 0} x \ln(x + O(x^3)) = \lim_{x \to 0} (x\ln x + x\ln(1+O(x^2))) = 0\)</span>. Therefore <span>\(\lim_{x \to 0}(\sin x)^x = e^0 = 1\)</span>.</p>
Correct Answer: A