Understanding the Limit of x-1/x as x Approaches Infinity

When dealing with the behavior of functions as x approaches infinity, understanding the limit of x-1/x becomes crucial. This article delves into the mathematical intricacies and provides clarity on the limit, the range of the function, and the derivative method to find critical points.

Understanding the Limit

The limit of the expression x-1/x as x approaches infinity can be evaluated by simplifying the expression. Let's start with the given expression:

Let Q x - 1/x. As x gets larger, the term 1/x approaches 0, making x - 1/x approximately equal to x. Therefore, the limit of Q as x approaches infinity is:

lim (x - 1/x)  lim x  infinity

Similarly, if you meant R x / (1/x) (or x * x^(-1)), then as x gets very large, the term 1/x fades toward zero, and the expression x / (1/x) approaches infinity, leading to:

lim (x * 1/x)  lim x  infinity

Therefore, the limit of the expression x - 1/x or x * 1/x as x approaches infinity is infinity.

Range of the Function y x - 1/x

The range of the function y x - 1/x is given by the intervals [-∞, -2] U [2, ∞]. This can be understood by analyzing the behavior of the function and its critical points.

Derivative Method for Finding Critical Points

To find the critical points of the function y x - 1/x, we can use the derivative method:

f(x)  x - 1/xf'(x)  1 - 1/x^2

Solving for f'(x) 0, we get:

1 - 1/x^2  ^2 - 1    pm;1

To determine whether these points are maxima or minima, we use the second derivative test:

f''(x)  2/x^3

Evaluating the second derivative at the critical points:

f''(-1)  -2 (indicating a local maximum)f''(1)  2 (indicating a local minimum)

Thus, the function has a local maximum of y -2 at x -1 and a local minimum of y 2 at x 1.

Behavior Based on the Sign of x

Case 1: Positive x Using the Arithmetic Mean-Geometric Mean Inequality (AM-GM) for positive x, we can find the maximum and minimum values:

AM (x 1/x)/2, GM 1 Since AM ge; GM, we have:

AM (x 1/x)/2 ge; 1

Solving for x, we get:

x 1/x ge; 2

x - 1/x le; 2

Therefore, the maximum value of x - 1/x for positive x is 2.

Case 2: Negative x Similarly, for negative x, we can use the AM-GM inequality:

AM (-x -1/x)/2, GM 1 Since AM ge; GM, we have:

AM (-x -1/x)/2 ge; 1

Solving for x, we get:

-x - 1/x le; -2

x - 1/x ge; -2

Therefore, the minimum value of x - 1/x for negative x is -2.

Combining both cases, the function y x - 1/x ranges from [-∞, -2] U [2, ∞].

Conclusion

The function y x - 1/x is discontinuous at x 0. However, by analyzing its behavior and critical points, we can determine its range and local maxima and minima. The limit of the expression as x approaches infinity is infinity, and the function ranges from [-∞, -2] U [2, ∞].