Limits, Continuity & Differentiability
Logarithmic limit properties
Grade 12
Question:
<p><strong>Ex. 40 (C):</strong> <span class="math">\lim_{x \to 0^+} (\ln \sin 3x - \ln(x^4 + ex^3))\</span> equals</p>
Step-by-Step Solution
Key Concept: Combine logarithms first, then analyze the ratio inside using Taylor series or L'Hôpital's rule.
<p><strong>Solution:</strong> Using properties of logarithms and L'Hôpital's rule:</p><p>$$\lim_{x \to 0^+} \ln\left(\frac{\sin 3x}{x^4 + ex^3}\right) = -1$$</p><p>∴ Answer is (q)</p>
Correct Answer: q