Limits, Continuity & Differentiability
Limits involving integrals
Grade 12
Question:
<p>\(\lim_{x \to 0} \frac{\int_0^x \frac{e^{\sin(tx)}}{x} dt}{x}\) equals to:</p>
<p>(a) 1</p>
<p>(b) 2</p>
<p>(c) \(e\)</p>
<p>(d) Does not exist</p>
Step-by-Step Solution
Key Concept: Use L'Hôpital's rule and Leibniz integral rule for differentiation of parametric integrals to evaluate the limit.
<p><strong>Step 1:</strong> Apply L'Hôpital's rule since the limit has form \(\frac{0}{0}\).</p><p><strong>Step 2:</strong> Differentiate numerator using Leibniz rule: \(\frac{d}{dx}\int_0^x \frac{e^{\sin(tx)}}{x} dt = \frac{e^{\sin(x \cdot x)}}{x} \cdot x + \int_0^x \frac{\partial}{\partial x}\left(\frac{e^{\sin(tx)}}{x}\right) dt\).</p><p><strong>Step 3:</strong> Simplify: As \(x \to 0\), the inner integral vanishes and \(e^{\sin(x^2)} \to e^0 = 1\).</p><p><strong>Step 4:</strong> The denominator derivative is 1, so \(\lim_{x \to 0} \frac{e^{\sin(x^2)}}{1} = 1\). ∴ Answer is (a).</p>
Correct Answer: a