(2n) /(n )^2

Jasonbradley

3 min read Jun 16, 2024

Exploring the Function (2n) / (n^2)

The expression (2n) / (n^2) represents a simple rational function. Let's explore its properties and behavior.

Simplifying the Expression

First, we can simplify the expression by canceling out a common factor of 'n' in the numerator and denominator:

(2n) / (n^2) = 2 / n

This simplified form clearly shows that the function is undefined when n = 0, as division by zero is not allowed.

Analyzing the Function's Behavior

As n approaches positive infinity (n → ∞), the function approaches zero. This is because the denominator grows much faster than the numerator, making the fraction progressively smaller.
As n approaches negative infinity (n → -∞), the function also approaches zero. The same principle applies, with the denominator growing much faster.
As n approaches zero from the positive side (n → 0+), the function approaches positive infinity. The denominator becomes extremely small, making the fraction extremely large.
As n approaches zero from the negative side (n → 0-), the function approaches negative infinity. The denominator becomes extremely small, but negative, making the fraction extremely large and negative.

Graphing the Function

The graph of the function (2/n) is a hyperbola. It has two branches: one in the first quadrant (n > 0) and one in the third quadrant (n < 0). The function is asymptotic to the x-axis (y = 0), and it has a vertical asymptote at n = 0.

Applications

The function (2n) / (n^2) can be used in various mathematical contexts, including:

Modeling physical phenomena: Some physical processes can be modeled using functions with similar behavior to this one.
Calculus: The function can be used as an example for studying limits, continuity, and derivatives.
Computer science: The function can be used in algorithms and data structures.

Conclusion

The function (2n) / (n^2), or its simplified form (2/n), is a simple yet insightful example of a rational function. Its behavior, characterized by its asymptotes and its approaching zero at infinity, highlights important concepts in mathematics and can be applied in various fields.