Limits, Continuity & Differentiability
Limits with Special Functions
Grade 12
Question:
<p>The value of $\lim_{x \to 0} \frac{\tan(\{1/x\}) \sin(ax)}{\sin(bx)}$, where $\{x\}$ denotes the fractional part function is</p>
<p>(a) 1</p>
<p>(b) $\tan 1$</p>
<p>(c) $\sin 1$</p>
<p>(d) Doesn't exist</p>
Step-by-Step Solution
Key Concept: Recognize that the fractional part function {1/x} oscillates in [0,1) and use the standard limit for sine ratios.
<p><strong>Solution:</strong> As $x \to 0$, we have $\frac{1}{x} \to \infty$, so the fractional part $\{1/x\}$ oscillates between 0 and 1. However, $\sin(ax) \to 0$ and $\sin(bx) \to 0$ as $x \to 0$.</p><p>Using standard limit: $\lim_{x \to 0} \frac{\sin(ax)}{\sin(bx)} = \frac{a}{b}$</p><p>Since $\{1/x\}$ is bounded between 0 and 1, $\tan(\{1/x\})$ is bounded. The limit becomes:</p><p>$\lim_{x \to 0} \tan(\{1/x\}) \cdot \frac{\sin(ax)}{\sin(bx)} = \tan(1) \cdot \frac{a}{b}$</p><p>∴ Answer is (b) $\tan 1$</p>
Correct Answer: b