Let's go over the questions one by one:

1. $\lim_{x \to 2} \frac{1 - \frac{2}{x}}{x^2 - x - 2}$

First, we factor the denominator: $x^2 - x - 2 = (x - 2)(x + 1)$

Next, simplify the numerator: $1 - \frac{2}{x} = \frac{x - 2}{x}$

Thus, the expression becomes: $\lim_{x \to 2} \frac{\frac{x - 2}{x}}{(x - 2)(x + 1)}$ Canceling $(x - 2)$ from numerator and denominator: $\lim_{x \to 2} \frac{1}{x(x + 1)}$ Substituting $x = 2$: $\frac{1}{2(2 + 1)} = \frac{1}{6}$ Answer: $\frac{1}{6}$

2. $\lim_{x \to 0} \frac{4 - 2x + x^2}{2 + x^2}$

Substitute $x = 0$ directly: $\frac{4 - 2(0) + (0)^2}{2 + (0)^2} = \frac{4}{2} = 2$ Answer: $2$

3. $\lim_{x \to 0} \frac{1 - \cos(2x)}{x \tan(x)}$

First, recall that $\cos(2x) \approx 1 - 2x^2$ for small $x$, and $\tan(x) \approx x$ for small $x$.

This simplifies the expression to: $\frac{1 - (1 - 2x^2)}{x \cdot x} = \frac{2x^2}{x^2} = 2$ Answer: $2$

4. Derivatives

a. $\text{cot}(x)$

The derivative of $\text{cot}(x)$ is: $\frac{d}{dx}[\text{cot}(x)] = -\csc^2(x)$

b. $\cos(x) \tan^2(x)$

Use the product rule: [ \frac{d}{dx} [\cos(x) \tan^2(x)] = \frac{d}{dx} [\cos(x)] \cdot \tan^2(x) + \cos(x) \cdot \frac{d}{dx} [\tan^2(x)] ] $= -\sin(x) \cdot \tan^2(x) + \cos(x) \cdot 2\tan(x) \sec^2(x)$

c. $\cot^2(x)$

Using the chain rule: [ \frac{d}{dx} [\cot^2(x)] = 2\cot(x) \cdot \frac{d}{dx} [\cot(x)] = 2\cot(x)(-\csc^2(x)) = -2\cot(x)\csc^2(x) ]

d. $\sec^2(x)$

The derivative of $\sec^2(x)$ is: $\frac{d}{dx}[\sec^2(x)] = 2\sec(x) \cdot \sec(x) \tan(x) = 2\sec^2(x) \tan(x)$

e. $\sqrt{x - x\sin^2(x)}$

Use the chain rule and simplify: $\frac{d}{dx} \left( \sqrt{x - x \sin^2(x)} \right) = \frac{1}{2\sqrt{x - x \sin^2(x)}} \cdot \left( 1 - (\sin^2(x) + 2x\sin(x)\cos(x)) \right)$ This requires further simplification, but the general form follows the chain rule.

Would you like any further details or explanations?

Follow-up questions:

1. How does the limit of $\frac{1 - \cos(2x)}{x\tan(x)}$ change with different trigonometric approximations?
2. Could you expand on how the chain rule applies to $\cot^2(x)$?
3. Why is it necessary to use the product rule for $\cos(x)\tan^2(x)$?
4. What happens if the denominator in the first limit cannot be factored?
5. How do small-angle approximations help in solving limits like in question 3?

Tip: Always check for factorization or small-angle approximations when handling limits with trigonometric expressions.