To determine if a function approaches zero as $x$ approaches infinity, you can analyze its behavior using a few methods:

1. Limit Calculation: Compute the limit of the function as $x$ approaches infinity. If $\lim_{x \to \infty} f(x) = 0$, then the function approaches zero.

2. End Behavior Analysis: Look at the dominant term in the function (especially in polynomial functions). The term with the highest degree often dictates the behavior as $x$ goes to infinity.

3. Asymptotic Analysis: For rational functions, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the denominator, the function approaches zero.

4. Graphical Analysis: Plot the function to visually inspect its behavior as $x$ increases.

5. Growth Rates: Examine whether the function has exponential, polynomial, or logarithmic growth. Functions like $\frac{1}{x^n}$ (for $n > 0$) approach zero as $x$ goes to infinity.

If you have a specific function in mind, I can help analyze it further! Would you like to explore that?

Follow-up Questions:

1. What specific function are you interested in analyzing?
2. Do you want to see examples of different types of functions?
3. Would you like to learn about limits in more detail?
4. Are you familiar with the concept of asymptotes?
5. Would you like to see a graphical representation of function behavior?

Tip: Always consider the degree of polynomials or the growth rates of functions when analyzing their limits at infinity.