Question

Assertion (A): $\lim _{x \rightarrow \pi / 2} \frac{\sin \left(\cot ^{2} x\right)}{(\pi-2 x)^{2}}=\frac{1}{2}$ Reason $(\mathbf{R}): \lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1\( and \)\lim _{\theta \rightarrow 0} \frac{\tan \theta}{\theta}=1\(, where \)\theta$ is measured in radians.

Short answer

The limit of the given function is \(4\).

Step-by-step solution

Identify Variables and Functions

Let \(A(x) = \cot^2x\) and \(B(x) = (\pi - 2x)^2\). Now we have to find \(\lim_{x\rightarrow\pi/2}\frac{\sin A(x)}{B(x)}\).

Rewrite Cotangent in Terms of Sine and Cosine

Recall that \(\cot{x} = \frac {\cos{x}}{\sin{x}}\), so \(\cot^2x = \frac {\cos^2{x}}{\sin^2{x}}\). Now we have $$A(x) = \frac {\cos^2{x}}{\sin^2{x}}$$

Substitution and Simplification

Let \(y = \pi - 2x\). Then as \(x \rightarrow \pi/2\), we have \(y \rightarrow 0\). So, we can rewrite the limit as: $$\lim_{y\rightarrow0}\frac{\sin\left(\frac{\cos^2(\frac{\pi - y}{2})}{\sin^2(\frac{\pi - y}{2})}\right)}{y^2}$$

Apply the Sine Double Angle Formula

Recall that \(\sin(\pi - \theta) = \sin{\theta}\). Therefore, $$\lim_{y\rightarrow0}\frac{\sin\left(\frac{\cos^2(\frac{y}{2})}{\sin^2(\frac{y}{2})}\right)}{y^2}$$

Apply the Given Limits

Applying the given limits: $$\lim_{\theta\rightarrow0}\frac{\sin\theta}{\theta}=1 \text{ and } \lim_{\theta\rightarrow0}\frac{\tan\theta}{\theta}=1$$ Substitute \(\theta = \frac{\cos^2(\frac{y}{2})}{\sin^2(\frac{y}{2})}\) in the first limit, and \(\theta = \frac{y}{2}\) in the second limit. Then we can rewrite the expression as: $$\lim_{y\rightarrow0}\frac{\frac{\sin\left(\frac{\cos^2(\frac{y}{2})}{\sin^2(\frac{y}{2})}\right)}{\frac{\cos^2(\frac{y}{2})}{\sin^2(\frac{y}{2})}}}{\frac{y^2}{\frac{y^2}{4}}}$$

Simplify the Expression

Simplify the expression: $$\lim_{y\rightarrow0}\frac{4\sin\left(\frac{\cos^2(\frac{y}{2})}{\sin^2(\frac{y}{2})}\right)}{y^2}$$ Now, use the limit result from Step 5: $$\frac{4}{\lim_{y\rightarrow0}\frac{y^2}{\frac{y^2}{4}}} = \frac{4}{1} = 4$$

Apply the Limit Product Rule

Apply the limit product rule, with \(C(y) = \frac{\sin\left(\frac{\cos^2(\frac{y}{2})}{\sin^2(\frac{y}{2})}\right)}{\frac{\cos^2(\frac{y}{2})}{\sin^2(\frac{y}{2})}}\) and \(D(y) = \frac{y^2}{\frac{y^2}{4}}\): $$\lim_{y\rightarrow0}C(y) \times \lim_{y\rightarrow0}D(y) = \frac{1}{2}$$ Given our analysis and solution, Assertion (A) is incorrect as it claims the limit to be \(\frac{1}{2}\) when the limit equals to \(4\) according to our calculations.