https://file.aiquickdraw.com/mathbot/file/172707397806861pytttb.jpg

The expression in the image is:

\[
\lim_{{x \to 0}} \frac{{3 + x^2}}{{x}}
\]

Let's solve this step by step.

### Step 1: Direct Substitution
If we directly substitute \( x = 0 \) into the expression, we get:

\[
\frac{{3 + 0^2}}{{0}} = \frac{3}{0}
\]

This results in division by zero, which suggests the limit does not exist in the ordinary sense. However, to determine if the limit approaches \( +\infty \) or \( -\infty \), we must examine the behavior of the function as \( x \) approaches zero from both the positive and negative directions.

### Step 2: Analyze the behavior from both sides

- **As \( x \to 0^+ \)** (from the right side):
  \[
  \frac{{3 + x^2}}{{x}} = \frac{3 + x^2}{x}
  \]
  For small positive \( x \), \( x > 0 \), and the denominator \( x \) is positive. So the expression tends to \( +\infty \).

- **As \( x \to 0^- \)** (from the left side):
  \[
  \frac{{3 + x^2}}{{x}} = \frac{3 + x^2}{x}
  \]
  For small negative \( x \), \( x < 0 \), and the denominator is negative. So the expression tends to \( -\infty \).

### Conclusion:
Since the limit approaches \( +\infty \) from the right and \( -\infty \) from the left, the overall limit does **not exist** because the one-sided limits are not equal.

Would you like more details on any of the steps or have further questions?

Here are some related questions you might find helpful:
1. What are one-sided limits, and how are they evaluated?
2. Can limits that tend to infinity still be considered defined in any context?
3. How do you handle limits involving division by zero?
4. What are some real-world applications of evaluating limits in calculus?
5. What are indeterminate forms, and how are they resolved using L'Hopital's Rule?

**Tip**: When limits approach infinity or negative infinity from different sides, it's essential to analyze each direction separately to fully understand the behavior.

Evaluate the limit as x approaches 0 for (3 + x^2) / x.

Explore the detailed solution to the limit of (3 + x^2) / x as x approaches 0. This problem demonstrates the behavior of the function near zero, involving one-sided limits and division by zero. Suitable for advanced high school or early college calculus students.

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