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.