Limits, Continuity & Differentiability
Evaluation of Limits using Standard Forms
Grade 12
Question:
<p><strong>Ex. 36:</strong> Statement I: $\lim_{x \to 3/2} \frac{\sin(\cot^2 x)}{(3-2x)^2} = \frac{1}{2}$</p><p>Statement II: $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$ and $\lim_{\theta \to 0} \frac{\tan \theta}{\theta} = 1$, where $\theta$ is measured in radians.</p>
<p>(a) Statement I is true, Statement II is true; Statement II is correct explanation for Statement I</p>
<p>(b) Statement I is true, Statement II is true; Statement II is not the correct explanation for Statement I</p>
<p>(c) Statement I is true, Statement II is false</p>
<p>(d) Statement I is false, Statement II is true</p>
Step-by-Step Solution
Key Concept: Recognize standard trigonometric limits and apply them carefully to evaluate indeterminate forms; verify both statements independently.
<p><strong>Solution:</strong> We evaluate the limit in Statement I. As $x \to 3/2$, we have $\cot 2x \to \cot 3 \to 0$, so $\sin(\cot^2 x) \to \sin(0) = 0$ and $(3-2x)^2 \to 0$. This is a $\frac{0}{0}$ form. Using the standard limits in Statement II and L'Hôpital's rule or series expansion, the limit evaluates to a different value than $\frac{1}{2}$. Statement II contains correct standard limits in trigonometry.</p><p>∴ Answer is (d): Statement I is false, Statement II is true.</p>
Correct Answer: D