It seems you've shared an image containing a variety of limit problems. I'll guide you through solving each one. Here is a step-by-step approach to the first few limits; let me know if you'd like the solutions to more of these, as they involve distinct techniques.

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

To solve this limit, let's analyze the behavior of the function as $x \to 1$ by simplifying the expression:

• First, observe that $x^2 - 1 = (x - 1)(x + 1)$.
• We need to consider the absolute value $|x^2 - 1|$, which will depend on whether $x \to 1^+$ (from the right) or $x \to 1^-$ (from the left).

Breaking it into left and right limits might simplify the calculation, as this allows us to handle $|x^2 - 1|$ appropriately. Please let me know if you would like a detailed breakdown.

2. $\lim_{x \to +\infty} 2x \sqrt{\frac{x}{x - 2}}$

As $x \to +\infty$:

• The term inside the square root, $\frac{x}{x - 2}$, tends to 1 as $x$ grows very large.
• So, we can simplify by approximating $\sqrt{\frac{x}{x - 2}} \approx 1$.

With this approach, you’ll be able to find the behavior of the expression as $x \to +\infty$.

3. $\lim_{x \to -\infty} \frac{x^3 + 4x^2 + 3x - 1}{(x + 1)^2}$

As $x \to -\infty$:

• The leading terms in both the numerator and the denominator dominate the behavior of the function.
• Simplifying by dividing by the highest power of $x$ will help us evaluate this limit.

4. $\lim_{x \to 0} \frac{\sin(-2x)}{3x}$

Using the standard limit $\lim_{x \to 0} \frac{\sin(kx)}{kx} = 1$:

• We can simplify this expression by factoring out constants, allowing us to find the limit.

5. $\lim_{x \to -6} \frac{3 - \sqrt{x + 3}}{x + 6}$

This limit requires rationalizing the expression to simplify the square root term in the numerator. By multiplying the numerator and denominator by the conjugate, you can proceed with the calculation.

Would you like a full solution for each of these, or for specific ones? Here are related questions to expand your understanding:

1. How does one handle absolute values in limits approaching specific points?
2. Why is rationalizing helpful in limits involving square roots?
3. What are common techniques for handling infinity limits in polynomial and radical expressions?
4. How do trigonometric limits leverage known standard limits, such as $\frac{\sin(x)}{x}$?
5. What’s the role of the conjugate in simplifying complex expressions?

Tip: When dealing with limits involving infinity, identify and focus on the highest degree terms, as they tend to dominate the behavior of the function at large values of $x$.