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The problem provided is:

\[
\lim_{{x \to 3^+}} \left( 3 + \frac{{2x}}{{x - 3}} \right)
\]

We are asked to evaluate this limit as \( x \) approaches 3 from the right-hand side (\( x \to 3^+ \)).

### Step-by-Step Solution:

1. **Substitute \( x = 3 \) directly (as a first check):**
   \[
   3 + \frac{{2(3)}}{{3 - 3}} = 3 + \frac{6}{0}
   \]
   Since division by zero occurs, this indicates a **vertical asymptote** and suggests that the limit might approach either positive or negative infinity.

2. **Examine the behavior as \( x \to 3^+ \):**

   For values of \( x \) slightly greater than 3 (from the right), the denominator \( x - 3 \) is a small positive number, while the numerator \( 2x \) is approximately 6.
   
   Therefore, as \( x \to 3^+ \), we have:
   \[
   \frac{2x}{x - 3} \to \frac{6}{+0^+} = +\infty
   \]
   This means the fraction grows without bound as \( x \) approaches 3 from the right.

3. **Conclusion:**
   \[
   \lim_{{x \to 3^+}} \left( 3 + \frac{{2x}}{{x - 3}} \right) = 3 + (+\infty) = +\infty
   \]

Thus, the limit is \( +\infty \).

Would you like further clarification or details on this solution?

Here are some related questions to explore further:
1. What happens if we evaluate the same limit from the left (\( x \to 3^- \))?
2. How do vertical asymptotes affect the behavior of limits?
3. Can this function be simplified or transformed for easier limit evaluation?
4. How would you compute limits approaching infinity or negative infinity?
5. How do limits relate to the concept of continuity at a point?

**Tip:** When dealing with limits involving division by a small number near zero, it's crucial to examine whether that small number is positive or negative to determine if the limit heads towards positive or negative infinity.

This problem evaluates the limit lim(x → 3+) (3 + 2x/(x - 3)), illustrating the concept of vertical asymptotes and infinite limits. The solution discusses why the limit approaches positive infinity as x nears 3 from the right. Suitable for advanced high school calculus students.

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