It looks like you've uploaded an image containing limit problems. I will transcribe and solve each of the limits step by step.

Problem 5.1

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

Solution:

For large values of $x$, the higher degree terms dominate. So, we approximate the expression by considering the dominant terms:

$\lim_{x \to \infty} \frac{2 + 3x}{x^2 + 1} \approx \lim_{x \to \infty} \frac{3x}{x^2}$

This simplifies to:

$\lim_{x \to \infty} \frac{3}{x} = 0$

Thus, the limit is: $\boxed{0}$

Problem 5.2

$\lim_{x \to 2^-} \frac{x^3 - |x|}{x^3 - 2x^2 - x + 2}$

Solution:

First, observe that as $x \to 2^-$, we approach from the left, so for values of $x$ near 2, we simplify the expression.

Let's directly substitute $x = 2$:

For $|x|$ (since $x$ is approaching from the left, $x < 0$):

For now, substituting $x = 2$ in, and checking for undefined behavior.

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