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.

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.

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.

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.