Lim x tends infinity x log (1+(x/2)-log(x/2))

We are tasked with evaluating the limit of the expression:

\[
\lim_{x \to \infty} x \log\left(1 + \frac{x}{2} - \log\left(\frac{x}{2}\right)\right)
\]

### Step-by-Step Solution:

1. **Simplify the argument inside the logarithm:**

   \[
   \log\left(1 + \frac{x}{2} - \log\left(\frac{x}{2}\right)\right)
   \]

   For large \( x \), we know that \( \log\left(\frac{x}{2}\right) \approx \log(x) - \log(2) \). Hence, the expression inside the logarithm can be approximated by:

   \[
   1 + \frac{x}{2} - \log(x) + \log(2)
   \]

   For large \( x \), the term \( \frac{x}{2} \) dominates the expression, so we approximate the expression as:

   \[
   \frac{x}{2}
   \]

2. **Take the logarithm:**

   The logarithm of \( \frac{x}{2} \) is:

   \[
   \log\left(\frac{x}{2}\right) = \log(x) - \log(2)
   \]

3. **Multiply by \( x \):**

   Now we multiply the logarithmic expression by \( x \):

   \[
   x \log\left(\frac{x}{2}\right) = x \left(\log(x) - \log(2)\right)
   \]

   This simplifies to:

   \[
   x \log(x) - x \log(2)
   \]

4. **Take the limit:**

   As \( x \to \infty \), the dominant term is \( x \log(x) \), and the term \( -x \log(2) \) is insignificant in comparison. Therefore, the limit of the original expression is dominated by the term \( x \log(x) \).

   Thus, the limit is:

   \[
   \lim_{x \to \infty} \left( x \log(x) - x \log(2) \right) = \infty
   \]

### Final Answer:

\[
\lim_{x \to \infty} x \log\left(1 + \frac{x}{2} - \log\left(\frac{x}{2}\right)\right) = \infty
\]

Would you like more details or have any further questions?

Here are 5 related questions for further exploration:
1. How does logarithmic growth compare with polynomial growth?
2. What is the asymptotic behavior of the function \( x \log(x) \)?
3. How do you apply L'Hopital's Rule to limits with logarithms?
4. What are the properties of logarithms that simplify limit calculations?
5. How do you find limits at infinity for functions involving both polynomial and logarithmic terms?

**Tip:** When dealing with limits involving logarithms and polynomials, always identify the dominant term as \( x \to \infty \) to simplify the expression.

Explore the solution for the limit of x log(1 + x/2 - log(x/2)) as x approaches infinity. This problem involves logarithmic functions and asymptotic analysis, with detailed steps on simplifying the expression and determining the dominant term.

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