Let's evaluate the three limits one by one. Here are the step-by-step solutions:

1. Limit 1: $\lim_{x \to 0^-} \frac{3x + 4}{x^2}$

• For $x \to 0^-$, the denominator $x^2$ remains positive because it is squared, but the numerator depends on $x$.
• When $x \to 0^-$, $3x$ becomes slightly negative, and $4$ remains positive.
• Therefore, $3x + 4 \to 4$ (slightly less than 4 from the left).

So,
$\lim_{x \to 0^-} \frac{3x + 4}{x^2} = \frac{\text{small negative}}{\text{small positive}} \to -\infty.$

Conclusion:
$\lim_{x \to 0^-} \frac{3x + 4}{x^2} = -\infty.$

2. Limit 2: $\lim_{x \to +\infty} \frac{3 - 2x^2}{x^2 + 1}$

• The dominant term in both the numerator and denominator is $x^2$ as $x \to +\infty$.
• Divide both numerator and denominator by $x^2$ to simplify:

$\frac{3 - 2x^2}{x^2 + 1} = \frac{\frac{3}{x^2} - 2}{1 + \frac{1}{x^2}}.$

• As $x \to +\infty$, terms involving $\frac{1}{x^2} \to 0$, so the expression simplifies to:

$\frac{0 - 2}{1 + 0} = -2.$

Conclusion:
$\lim_{x \to +\infty} \frac{3 - 2x^2}{x^2 + 1} = -2.$

3. Limit 3: $\lim_{x \to -\infty} \frac{x + 1}{x^3 - 1}$

• For $x \to -\infty$, the dominant term in the numerator is $x$, and in the denominator, $x^3$ dominates.
• Divide both numerator and denominator by $x^3$:

$\frac{x + 1}{x^3 - 1} = \frac{\frac{x}{x^3} + \frac{1}{x^3}}{1 - \frac{1}{x^3}}.$

Simplify each term:

• $\frac{x}{x^3} = \frac{1}{x^2} \to 0$ as $x \to -\infty$.
• $\frac{1}{x^3} \to 0$ as $x \to -\infty$.

Thus, the limit simplifies to:

$\frac{0 + 0}{1 - 0} = 0.$

Conclusion:
$\lim_{x \to -\infty} \frac{x + 1}{x^3 - 1} = 0.$

Final Answers:

1. $\lim_{x \to 0^-} \frac{3x + 4}{x^2} = -\infty$
2. $\lim_{x \to +\infty} \frac{3 - 2x^2}{x^2 + 1} = -2$
3. $\lim_{x \to -\infty} \frac{x + 1}{x^3 - 1} = 0$

Do you need further details or have any additional questions?

Here are 5 related questions for practice:

1. Evaluate $\lim_{x \to \infty} \frac{5x^3 - x}{2x^3 + 3x}$.
2. Find $\lim_{x \to 0^+} \frac{2x + 1}{x^2}$.
3. Calculate $\lim_{x \to -\infty} \frac{x^2 - 3x}{x^2 + 2}$.
4. Evaluate $\lim_{x \to \infty} \frac{4x^2 - x + 5}{x^2 + 7}$.
5. Find $\lim_{x \to 0} \frac{\sin x}{x}$.

Tip:

When evaluating limits involving large $x$, always focus on the dominant terms (highest powers of $x$) in the numerator and denominator.