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The image you provided contains a mathematical solution related to limits and indeterminate forms, written in Arabic. I will break it down step by step, translating and explaining the key concepts.

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### **Explanation of the Image:**

#### 1. **Top Right - عنوان: "لتحدد النهاية"**
   This translates to: **"To find the limit"**

   The task here is to find the limit of a function as \( x \to \frac{1}{2} \).

#### 2. **Top Limit Equation:**

   \[
   \lim_{x \to \frac{1}{2}} \frac{x + 3}{1 - 2x}
   \]

   This is a limit problem, and we need to evaluate it by substituting \( x = \frac{1}{2} \) into the expression. However, we see from the next steps that the function \( 1 - 2x \) goes to 0 at \( x = \frac{1}{2} \), leading to a potential indeterminate form \( \frac{0}{0} \).

#### 3. **Simplified Calculation:**
   - The image shows:
   
     \[
     \lim_{x \to \frac{1}{2}} (1 - 2x) = 0
     \]
     \[
     \lim_{x \to \frac{1}{2}} (x + 3) = \frac{7}{2}
     \]

   So, at \( x = \frac{1}{2} \), \( 1 - 2x \) approaches 0, and \( x + 3 \) approaches \( \frac{7}{2} \).

#### 4. **Behavior Around \( x = \frac{1}{2} \):**
   - The next section uses a table to evaluate the sign of \( 1 - 2x \) as \( x \) approaches \( \frac{1}{2} \) from both sides:
     - As \( x \to \frac{1}{2}^- \) (from the left), \( 1 - 2x \to +0 \)
     - As \( x \to \frac{1}{2}^+ \) (from the right), \( 1 - 2x \to -0 \)

     This shows that the function's behavior changes around \( x = \frac{1}{2} \).

#### 5. **Final Conclusion:**
   The final limit for \( \lim_{x \to \frac{1}{2}} \frac{x + 3}{1 - 2x} \) is calculated as follows:
   
   \[
   \lim_{x \to \frac{1}{2}^-} = +\infty
   \]
   \[
   \lim_{x \to \frac{1}{2}^+} = -\infty
   \]

   Hence, the limit does not exist as the function tends to \( +\infty \) from the left and \( -\infty \) from the right.

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### **Summary:**
- The limit problem \( \lim_{x \to \frac{1}{2}} \frac{x + 3}{1 - 2x} \) results in a vertical asymptote at \( x = \frac{1}{2} \).
- The behavior of the function differs as \( x \) approaches \( \frac{1}{2} \) from the left (positive infinity) and from the right (negative infinity), indicating that the two-sided limit does not exist.

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Would you like further details or have specific questions about this? Here are five related questions that might help expand understanding:

1. How do you determine the behavior of limits when a function approaches infinity?
2. What happens when both the numerator and denominator tend to 0 in a limit problem?
3. Can a limit exist if the left-hand and right-hand limits are different?
4. How do you deal with indeterminate forms like \( \frac{0}{0} \) in limit problems?
5. What are vertical asymptotes, and how do they affect limits?

**Tip:** In problems involving limits where substitution leads to an indeterminate form (like \( \frac{0}{0} \)), try simplifying the expression or consider using L'Hopital's Rule if applicable.

Solve the limit problem involving indeterminate forms as x approaches 1/2

This page provides a detailed solution to the limit problem lim(x -> 1/2) (x + 3) / (1 - 2x), explaining key concepts such as indeterminate forms, vertical asymptotes, and limit behavior. Suitable for advanced high school students in grades 11-12.

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