The function you've provided is:

\[
y = \left( 1 + \frac{1}{x} \right)^x
\]

This is a well-known mathematical expression that resembles the definition of the constant \( e \), which is the base of natural logarithms. As \( x \to \infty \), this expression approaches \( e \), the Euler's number, which is approximately 2.71828.

### Analyzing the Behavior of the Function:
1. **Asymptotic Behavior**: 
   \[
   \lim_{x \to \infty} \left( 1 + \frac{1}{x} \right)^x = e
   \]
   
   As \( x \) becomes very large, the function approximates \( e \). This is because the expression is often used as a limit definition of \( e \).

2. **Behavior as \( x \to 0^+ \)**:  
   As \( x \) approaches zero from the positive side, the function behaves differently. The expression inside the parentheses, \( \left(1 + \frac{1}{x} \right) \), grows large, but \( x \) also shrinks, leading to a rapid growth in the expression. Thus, the function tends to infinity.

3. **For Negative Values of \( x \)**:  
   When \( x \) is negative, the function becomes more complex. For negative values of \( x \), \( \left( 1 + \frac{1}{x} \right) \) could become negative or undefined depending on the value of \( x \). The behavior in this region needs more careful consideration.

Would you like to explore further details, such as its derivative or specific behavior in certain ranges?

### Related Questions:
1. How would you derive this function \( y \) with respect to \( x \)?
2. What happens to the function as \( x \to -1 \)?
3. Can you express this function as a limit as \( x \to \infty \)?
4. How does the function behave for small positive values of \( x \)?
5. Can this function be applied in approximating \( e \) for large \( x \)?

**Tip**: Functions involving limits, like this one, are useful for approximating constants such as \( e \) and can help understand growth rates and exponential behavior.

Explore the behavior of the exponential function y = (1 + 1/x)^x, which approaches Euler's number e as x tends to infinity. This problem covers key calculus concepts, including limits, exponential functions, and the limit definition of e. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation