Limits, Continuity & Differentiability
L'Hôpital's rule with integral functions
Grade 12
Question:
<p>Find \(\displaystyle\lim_{x \to \infty} \frac{\displaystyle\int_0^x e^{t^2}\, dt}{\displaystyle\int_0^x e^{2t^2}\, dt}\).</p>
Step-by-Step Solution
Key Concept: Apply L'Hôpital's rule to the indeterminate form ∞/∞ by differentiating numerator and denominator using the Fundamental Theorem of Calculus, which converts integrals to their integrands.
<p><strong>Step 1:</strong> Verify the form. As x → ∞, both ∫₀ˣ e^(t²) dt and ∫₀ˣ e^(2t²) dt tend to ∞, giving the indeterminate form ∞/∞.</p><p><strong>Step 2:</strong> Apply L'Hôpital's rule. By the Fundamental Theorem of Calculus:</p><p>d/dx[∫₀ˣ e^(t²) dt] = e^(x²)</p><p>d/dx[∫₀ˣ e^(2t²) dt] = e^(2x²)</p><p><strong>Step 3:</strong> Evaluate the limit:</p><p>lim(x→∞) e^(x²)/e^(2x²) = lim(x→∞) e^(x² - 2x²) = lim(x→∞) e^(-x²) = 0</p><p><strong>Step 4:</strong> As x → ∞, the exponential e^(2x²) in the denominator grows much faster than e^(x²) in the numerator, causing the ratio to decay to 0.</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: 0