Let's go through each limit problem one by one.

Problem 1

The limit is given by: $\lim_{x \to x_0} \frac{x^2 - 3x + 2}{4 - x - 3x^2}$ with three different values for $x_0$:

• $x_0 = -1$
• $x_0 = 1$
• $x_0 = \infty$

For each case, we would substitute $x_0$ and simplify if possible.

Problem 2

The limit is: $\lim_{x \to 0} \frac{\sin 4x}{2x \cos 3x}$ This problem likely involves trigonometric identities or simplifications, especially around $\sin(x)/x = 1$ as $x \to 0$.

Problem 3

This limit is: $\lim_{x \to \infty} \left( \frac{3x + 2}{3x - 4} \right)^{2 - x}$ This limit might involve exponential decay as $x \to \infty$, depending on the behavior of the base.

Problem 4

The limit is: $\lim_{x \to 4} \frac{\sqrt{x + 12} - \sqrt{4 - x}}{x^2 + 2x - 8}$ This is a difference of square roots divided by a polynomial. We might need to rationalize the numerator or simplify the denominator.

Would you like a detailed solution for each part?

Follow-up Questions

1. How can limits involving infinity be evaluated more effectively?
2. What techniques help simplify complex fractions in limit problems?
3. How do trigonometric limits work, especially those using $\sin(x)/x$?
4. What strategies are useful for rationalizing square roots in limit problems?
5. When should L'Hôpital's Rule be applied in limit problems?

Tip

When facing limits that result in indeterminate forms, consider factoring, rationalizing, or using L'Hôpital's Rule to simplify.