Understanding the Limit of x/x as x Approaches Infinity

The concept of limits is a cornerstone in calculus, allowing us to analyze the behavior of functions as variables approach certain values. A particularly interesting case is the limit of the function (f(x) frac{x}{x}) as x approaches infinity.

Definition and Basic Properties

Define the function (f : mathbb{R}^* to mathbb{R}) as (f(x) frac{x}{x}) for all (x in mathbb{R}^*), where (mathbb{R}^* mathbb{R} setminus {0}). This definition simplifies to (f(x) 1) for all (x in mathbb{R}^*).

Limit Analysis

Consider a sequence (a_n in mathbb{R}^*) such that (a_n to infty) as (n to infty). We want to find the limit of the sequence (f(a_n)). Since (f(x) 1) for all (x in mathbb{R}^*), it follows that:

[f(a_n) 1 text{ for all } n in mathbb{N}]

Thus, the sequence (f(a_n)) is a constant sequence equal to 1, which converges to 1 as (n to infty). Therefore:

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

Alternative Perspectives

The limit of (frac{x}{x}) as (x to infty) can also be understood by breaking it into absolute terms:

[lim_{x to infty} frac{|x|}{x}]

As (x to infty), |(x)| equals (x). Therefore, the expression simplifies to:

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

Properties of the Function

The function (f(x)) can be further analyzed based on the sign of (x).

(x > 0): The function simplifies to (f(x) 1). (x The function simplifies to (f(x) -1). (x 0): The function is undefined since it involves division by zero.

Thus, the function is a constant function equal to 1 for positive values of (x) and -1 for negative values of (x), with a point of discontinuity at (x 0).

Conclusion

In conclusion, the limit of (frac{x}{x}) as (x to infty) is 1. This is due to the fact that the function is simplified to 1 for all positive values of (x), and the limit of a constant function as it approaches infinity is the constant itself.

For further reading and deeper understanding, you can explore more complex limits, infinite series, and real analysis concepts related to the behavior of functions as variables approach infinity.