What is the Limit of ( frac{x 1}{x} ) as ( x ) Approaches Infinity?

In this article, we will explore the behavior of the function ( frac{x 1}{x} ) as ( x ) approaches infinity. We will break down the algebraic manipulation of the function and provide a graphical interpretation to understand how it approaches its limit value.

Algebraic Approach

Let's start by examining the given function:

[ f(x) frac{x 1}{x} ]

To find the limit as ( x ) approaches infinity, we can simplify the expression:

[ f(x) frac{x 1}{x} frac{x}{x} frac{1}{x} 1 frac{1}{x} ]

Now, consider the limit:

[ lim_{x to infty} left( 1 frac{1}{x} right) ]

As ( x ) approaches infinity, ( frac{1}{x} ) approaches 0. Therefore, the expression simplifies to:

[ lim_{x to infty} left( 1 frac{1}{x} right) 1 0 1 ]

Graphical Interpretation

A graphical interpretation can also help us visualize this limit. When ( x ) is very large, the term ( frac{1}{x} ) becomes very small, close to zero. Let's look at a few values of ( x ) to see this:

If ( x 1000 ) The function ( frac{1000 1}{1000} frac{1001}{1000} 1.001 ) As ( x ) increases, the function gets closer to 1

The graph above shows that as ( x ) increases, the function ( f(x) ) approaches the horizontal asymptote at 1.

Properties of Limits

Using the properties of limits, we can break down the limit of ( frac{x 1}{x} ) as follows:

[ lim_{x to infty} left( 1 frac{1}{x} right) lim_{x to infty} 1 lim_{x to infty} frac{1}{x} ]

Since the limit of a constant is the constant itself, and the limit of ( frac{1}{x} ) as ( x ) approaches infinity is zero, we get:

[ lim_{x to infty} 1 lim_{x to infty} frac{1}{x} 1 0 1 ]

This confirms that the limit of ( frac{x 1}{x} ) as ( x ) approaches infinity is 1.

Real-World Applications

In terms of practical applications, the limit of ( frac{x 1}{x} ) to 1 is not extensively used in pragmatic scenarios. However, it is a fundamental concept in calculus and serves as a basis for understanding more complex limits and asymptotic behavior in mathematical and scientific fields.

For example, in economics, this concept can be helpful in modeling asymptotic behavior of economic functions. In computer science, it can be applied in analyzing the growth of algorithms as input sizes become infinitely large.

While the limit itself is not extensively utilized, it is a critical building block for more sophisticated mathematical and scientific analyses.