Understanding the Asymptotes of the Function ( f(x) frac{1}{x} )

Introduction

The function ( f(x) frac{1}{x} ) is a fundamental example in calculus, often introduced to illustrate the behavior of rational functions. This article delves into the asymptotic behavior of the function, specifically focusing on its vertical, horizontal, and oblique asymptotes. By analyzing these asymptotes, we can better understand the function's behavior at its boundaries and its long-term behavior as ( x ) approaches different values.

Identification of Asymptotes

Let's first identify the asymptotes of the function ( f(x) frac{1}{x} ). An asymptote is a line that a curve approaches but never touches. There are three types of asymptotes: vertical, horizontal, and oblique (slant).

Vertical Asymptote

Horizontal asymptotes are determined by the degrees of the numerator and denominator in a rational function. However, for the function ( f(x) frac{1}{x} ), we need to identify vertical asymptotes first, which occur where the function is undefined. In this case, the function is undefined when the denominator is zero:

1. Find where the expression ( frac{1}{x} ) is undefined.

x 0.

Since ( frac{1}{0} ) is undefined in mathematics, the function has a vertical asymptote at ( x 0 ).

Horizontal Asymptote

Next, we determine the horizontal asymptote by comparing the degrees of the numerator and the denominator. In the function ( f(x) frac{1}{x} ), the numerator is a constant (degree 0), and the denominator is ( x ) (degree 1):

2. If ( n ) is the degree of the numerator and ( m ) is the degree of the denominator, find n and m. n 0 m 1

Since ( n

No Oblique Asymptote

Oblique asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. In this case, since ( n 0 ) and ( m 1 ), there is no oblique asymptote:

3. If ( n ) is less than or equal to ( m ), there is no oblique asymptote.

No Oblique Asymptotes

Behavior of the Function at Key Points

To better understand the function's behavior, let's examine its behavior as ( x ) approaches key values.

As ( x ) Approaches Zero

As ( x ) approaches zero, the value of ( frac{1}{x} ) grows without bound. Specifically:

If ( x ) approaches zero from the right (positive values), then ( frac{1}{x} ) approaches positive infinity.

Example: 1/0.000001 1,000,000

If ( x ) approaches zero from the left (negative values), then ( frac{1}{x} ) approaches negative infinity.

Example: 1/(-0.000001) -1,000,000

As ( x ) Approaches ( pm infty )

As ( x ) approaches infinity or negative infinity, the value of ( frac{1}{x} ) approaches zero:

If ( x ) approaches infinity, ( frac{1}{x} ) approaches zero from the positive side.

If ( x ) approaches negative infinity, ( frac{1}{x} ) also approaches zero from the positive side.

Therefore, the function ( f(x) frac{1}{x} ) has horizontal asymptotes at ( y 0 ) and vertical asymptotes at ( x 0 ).

Summary

In summary, the function ( f(x) frac{1}{x} ) has vertical and horizontal asymptotes, but no oblique asymptotes. The vertical asymptote is at ( x 0 ), and the horizontal asymptote is at ( y 0 ). The function approaches zero as ( x ) grows large in either the positive or negative direction, and it grows without bound as ( x ) approaches zero from either direction.