Evaluate `lim_(xto pi//4) (1-cot^(3)x)/(2-cotx-cot^(3)x).`

Explanation

To evaluate the limit

\[ \lim_{x \to \frac{\pi}{4}} \frac{1 - \cot^3 x}{2 - \cot x - \cot^3 x}, \]

we can follow these steps:

### Step 1: Rewrite the expression
We can rewrite the numerator \(1 - \cot^3 x\) using the difference of cubes formula:

\[ 1 - \cot^3 x = (1 - \cot x)(1 + \cot x + \cot^2 x). \]

### Step 2: Rewrite the denominator
Next, we can rewrite the denominator \(2 - \cot x - \cot^3 x\). We can express \(2\) as \(1 + 1\):

\[ 2 - \cot x - \cot^3 x = (1 - \cot^3 x) + (1 - \cot x). \]

### Step 3: Substitute the rewritten forms
Now substituting the rewritten forms into the limit gives us:

\[ \lim_{x \to \frac{\pi}{4}} \frac{(1 - \cot x)(1 + \cot x + \cot^2 x)}{(1 - \cot^3 x) + (1 - \cot x)}. \]

### Step 4: Factor out common terms
Notice that \(1 - \cot^3 x\) can be factored as \( (1 - \cot x)(1 + \cot x + \cot^2 x) \). Thus, we can simplify:

\[ \lim_{x \to \frac{\pi}{4}} \frac{(1 - \cot x)(1 + \cot x + \cot^2 x)}{(1 - \cot x)(1 + \cot x + \cot^2 x) + (1 - \cot x)}. \]

### Step 5: Cancel out common factors
Since \(1 - \cot x\) is a common factor in both the numerator and the denominator, we can cancel it out (as long as \(x \neq \frac{\pi}{4}\)):

\[ \lim_{x \to \frac{\pi}{4}} \frac{1 + \cot x + \cot^2 x}{1 + \cot x + \cot^2 x + 1}. \]

### Step 6: Substitute \(x = \frac{\pi}{4}\)
Now we can substitute \(x = \frac{\pi}{4}\):

\[ \cot\left(\frac{\pi}{4}\right) = 1. \]

So we have:

\[ \lim_{x \to \frac{\pi}{4}} \frac{1 + 1 + 1^2}{1 + 1 + 1^2 + 1} = \frac{3}{4}. \]

### Final Answer
Thus, the limit evaluates to:

\[ \frac{3}{4}. \]