To evaluate the limit $\lim_{{x \to 0}} \frac{\sin(x)}{\sin^2(x) - 2\tan(x)}$, we need to simplify and analyze the expression as $x$ approaches 0.

First, let's recall the small-angle approximations for sine and tangent functions:

• $\sin(x) \approx x$
• $\tan(x) \approx x$

Let's rewrite the expression using these approximations: $\sin^2(x) \approx x^2 \quad \text{and} \quad 2\tan(x) \approx 2x$

So, the expression inside the limit becomes: $\frac{\sin(x)}{\sin^2(x) - 2\tan(x)} \approx \frac{x}{x^2 - 2x}$

Simplify the denominator: $x^2 - 2x = x(x - 2)$

Thus, the limit expression can be rewritten as: $\frac{x}{x(x - 2)} = \frac{x}{x^2 - 2x}$

We can cancel the common factor $x$ in the numerator and the denominator: $\frac{x}{x(x - 2)} = \frac{1}{x - 2}$

Now, we need to evaluate the limit: $\lim_{{x \to 0}} \frac{1}{x - 2}$

Substitute $x = 0$: $\frac{1}{0 - 2} = \frac{1}{-2} = -\frac{1}{2}$

Therefore, the limit is: $\boxed{-\frac{1}{2}}$

Do you have any questions or need more details?

Here are some related questions to further explore the topic:

1. What is the small-angle approximation for $\cos(x)$?
2. How does L'Hôpital's Rule apply to limits involving indeterminate forms?
3. Can you provide an example where L'Hôpital's Rule is necessary to evaluate a limit?
4. What are the different types of indeterminate forms?
5. How do Taylor series expansions help in evaluating limits?
6. What is the difference between $\lim_{{x \to 0}} \frac{\sin(x)}{x}$ and $\lim_{{x \to 0}} \frac{x}{\sin(x)}$?
7. How can we use series expansions to approximate $\tan(x)$ for small angles?
8. What is the importance of knowing the behavior of trigonometric functions near 0?

Tip: Always check if the limit results in an indeterminate form (like $\frac{0}{0}$ or $\frac{\infty}{\infty}$) before simplifying or applying L'Hôpital's Rule.