https://file.aiquickdraw.com/mathbot/file/17264924274198wjyslrl.jpg

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.

Let me know

Find the limits for the following expressions:

1. \( \lim_{x \to \infty} \frac{2 + 3x}{x^2 + 1} \)
2. \( \lim_{x \to 2^-} \frac{x^3 - |x|}{x^3 - 2x^2 - x + 2} \)
3. \( \lim_{x \to \infty} \frac{3x^3 + 2x + 1}{1 - \sqrt{x^2}} \)
4. \( \lim_{x \to 0} \left( \sqrt{x^2 + x} - \sqrt{x^2 - x} \right) \)

This page provides detailed solutions to a set of limit problems involving rational functions, absolute value expressions, and radical functions. Topics covered include handling infinite limits, piecewise functions, and square roots in limit calculations. Suitable for advanced high school or university-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation