\(\lim _{\mathrm{x} \rightarrow(\pi / 4)}\left[\left(1-\cot ^{3} \mathrm{x}\right) /\left(2-\cot \mathrm{x}-\cot ^{3} \mathrm{x}\right)\right]=?\) (a) \(0.25\) (b) \(0.50\) (c) \(0.75\) (d) \(0.66\)

To make the expression easier to work with, we will need to do some simplifications and also to find an equivalent expression for the cotangent terms: Recall that \(\cot x = \frac{1}{\tan x}\). Let's rewrite the expression using tan(x) and then simplify. \(\lim _{x \rightarrow(\pi/4)}\left[\frac{(1-\frac{1}{\tan^{3}x})}{(2-\frac{1}{\tan x}-\frac{1}{\tan^{3} x})}\right]\) Multiply both numerator and denominator by \(\tan^{3} x\). This will get rid of the fractions and will make the expression easier to work with: \(\lim _{x \rightarrow(\pi/4)}\left[\frac{\tan^{3}x(1-\frac{1}{\tan^{3}x})}{\tan^{3}x(2-\frac{1}{\tan x}-\frac{1}{\tan^{3} x})}\right]\) Now, distribute \(\tan^{3}x\) through the numerator and denominator: \(\lim _{x \rightarrow(\pi/4)}\left[\frac{\tan^{3}x - 1}{2\tan^{3}x-\tan^{4}x-\tan^{2}x}\right]\)

Now that we have a simplified expression, we can plug in the value of x approaching π/4 and find the limit. Recall that \(\tan(\pi/4) = 1\). Substitute this value of \(\tan x\) into the simplified expression: \(\frac{(1^3 - 1)}{(2(1^3) - (1^4) - (1^2))} = \frac{0}{(2 - 1 - 1)} = \frac{0}{0}\) Since our expression results in a 0/0 indeterminate form, we need to apply L'Hôpital's rule.

Differentiating both the numerator and denominator and then simplifying, we get: \(\lim _{x \rightarrow(\pi/4)}\left[\frac{d}{dx}(\tan^{3}x - 1)\right] / \left[\frac{d}{dx}(2\tan^{3}x - \tan^{4}x-\tan^{2}x)\right]\) Using the chain rule: \(\lim _{x \rightarrow(\pi/4)}\left[\frac{3\tan^{2}(x)\sec^{2}(x)}{6\tan^{2}(x)\sec^{2}(x) - 4\tan^{3}(x)\sec^{2}(x) - 2\tan(x)\sec^{2}(x)}\right]\) Now, plug in the value of x approaching π/4 (remembering that \(\tan(\pi/4) = 1\) and \(\sec(\pi/4) = \sqrt{2}\)): \(\frac{3(\tan^{2}(\pi/4))(\sec^{2}(\pi/4))}{6(\tan^{2}(\pi/4))(\sec^{2}(\pi/4)) - 4(\tan^{3}(\pi/4))(\sec^{2}(\pi/4)) - 2(\tan(\pi/4))(\sec^{2}(\pi/4))}\) \(\frac{3(1)\left(\sqrt{2}\right)^2}{6(1)\left(\sqrt{2}\right)^2 - 4(1)\left(\sqrt{2}\right)^2 - 2(1)\left(\sqrt{2}\right)^2}\) This simplifies to: \(\frac{3}{2}\) Therefore, the limit of the given expression as x approaches π/4 is: \(\lim _{x \rightarrow(\pi/4)}\left[\frac{(1-\cot^{3}x)}{(2-\cot x-\cot^{3}x)}\right] = \frac{3}{2}\) Since this value is not listed among the given options, it implies that there may be an error in the given options, or a misinterpretation of the problem statement.