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The question in the image asks which of the following statements is **false**, where \( x \) is a real number lying in the interval \( (0, 1) \), and it provides five options.

Let's evaluate each one:

1. **\( e^x < e \):**
   - Since \( x \in (0, 1) \), \( e^x \) will always be less than \( e \) because exponentiating a number less than 1 results in a value less than \( e \). So, this statement is **true**.

2. **\( e^{-x} < \frac{1}{e} \):**
   - Here \( e^{-x} = \frac{1}{e^x} \). Since \( x \in (0, 1) \), \( e^x \) is between \( 1 \) and \( e \), so \( \frac{1}{e^x} \) (which is \( e^{-x} \)) is between \( \frac{1}{e} \) and \( 1 \), meaning \( e^{-x} \) is greater than \( \frac{1}{e} \). Thus, the statement \( e^{-x} < \frac{1}{e} \) is **false**.

3. **\( \ln(x) < 0 \):**
   - Since \( x \in (0, 1) \), \( \ln(x) \) is negative because the logarithm of any number less than 1 is negative. So, this statement is **true**.

4. **\( e^{-x} < 1 \):**
   - Since \( e^{-x} = \frac{1}{e^x} \), and \( e^x > 1 \) for \( x \in (0, 1) \), this implies \( e^{-x} \) is indeed less than 1. So, this statement is **true**.

5. **\( \ln\left( \frac{1}{x} \right) > 0 \):**
   - Since \( x \in (0, 1) \), \( \frac{1}{x} > 1 \), and the logarithm of a number greater than 1 is positive. So, this statement is **true**.

### Conclusion:
The false statement is **\( e^{-x} < \frac{1}{e} \)**, which corresponds to the second option.

Would you like more details or explanations? Here are 5 related questions to expand the topic:

1. What is the behavior of the exponential function \( e^x \) when \( x \) is negative?
2. How does the logarithmic function behave for values in the interval \( (0, 1) \)?
3. Why is \( e^{-x} \) always greater than \( \frac{1}{e} \) for \( x \in (0, 1) \)?
4. How do exponential and logarithmic functions relate to each other in terms of inverse functions?
5. What are the properties of \( \ln(x) \) when \( x \to 0^+ \)?

### Tip:
When working with logarithmic inequalities, always keep in mind the domain restrictions to avoid undefined results (e.g., \( \ln(x) \) is only defined for \( x > 0 \)).

Let x be a real number lying in the interval (0, 1), Which of the following statements is false? (Recall that ln = log_e). The statements involve comparisons of exponential and logarithmic expressions.

Evaluate which statement involving exponential and logarithmic functions is false for x in the interval (0, 1). Understand key properties of e^x, e^{-x}, and ln(x) with detailed step-by-step explanations. Suitable for students in Grades 11-12.

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