Limits, Continuity & Differentiability
Indeterminate Forms
Grade 12
Question:
<p>The integer <span class="math inline">\(n\)</span> for which <span class="math inline">\(\lim_{x \to 0} (\sin x)^{1/x}\)</span> is a finite non-zero number, is [2002 AIEEE]</p>
<p>(a) 1</p>
<p>(b) 2</p>
<p>(c) 3</p>
<p>(d) 4</p>
Step-by-Step Solution
Key Concept: Convert the indeterminate form to an exponential and use standard limits and L'Hôpital's rule.
<p>Using the exponential form: <span class="math inline">$(\sin x)^{1/x} = e^{\frac{\ln(\sin x)}{x}}$</span>. As <span class="math inline">$x \to 0$</span>, we need <span class="math inline">$\lim_{x \to 0} \frac{\ln(\sin x)}{x}$</span> to be finite. Using L'Hôpital's rule and the fact that <span class="math inline">$\sin x \approx x$</span> for small <span class="math inline">$x$</span>, the value of <span class="math inline">$n = 1$</span> gives a finite non-zero limit.</p>
Correct Answer: A