Limits at Infinity and Horizontal Asymptotes

Introduction to Limits at Infinity

Limits at infinity describe the behavior of a function as the independent variable (usually x) increases or decreases without bound. These limits are essential for understanding the end behavior of functions and identifying horizontal asymptotes.

• Limit at Infinity: The value that f(x) approaches as x goes to positive or negative infinity.

• Horizontal Asymptote: A horizontal line y = c that the graph of f(x) approaches as x goes to infinity or negative infinity.

• Example: For f(x) = \frac{1}{x^2}, as x \to \pm\infty, f(x) \to 0.

Numerical Approach to Limits at Infinity

Numerically, we can observe the behavior of a function as x increases or decreases by evaluating the function at large values of x.

• As x becomes very large, f(x) approaches a constant value.

• Table of values can help visualize this behavior.

• Example: For f(x) = \frac{1}{x^2}, as x increases, f(x) gets closer to 0.

Definition: Limits at Infinity and Horizontal Asymptotes

Formal Definition

If f(x) becomes arbitrarily close to a finite number L for all sufficiently large and positive x, we write:

• The line y = L is a horizontal asymptote of f(x).

• Similarly, .

Using the Squeeze Theorem

The Squeeze Theorem can be used to evaluate limits at infinity, especially when the function is bounded between two simpler functions whose limits are known.

• Squeeze Theorem: If g(x) \leq f(x) \leq h(x) for all x in some interval, and , then .

• Example: For f(x) = \frac{5 \sin x}{x}, since and both bounds go to 0 as x \to \infty, .

Infinite Limits at Infinity

Definition and Behavior

If f(x) becomes arbitrarily large as x increases without bound, then . Infinite limits at infinity are often observed in polynomial and rational functions.

• Example: For f(x) = x^n (where n is a positive integer):

• If n is even, f(x) goes to infinity as x \to \pm\infty.

• If n is odd, f(x) goes to infinity as x \to \infty and negative infinity as x \to -\infty.

Reciprocals of Power Functions

The reciprocals of power functions, f(x) = \frac{1}{x^n}, approach zero as x increases or decreases without bound.

• Example: For f(x) = \frac{1}{x^2}, .

End Behavior of Rational and Algebraic Functions

Determining Limits at Infinity for Rational Functions

To determine the limit at infinity for a rational function f(x) = \frac{P(x)}{Q(x)}:

• Divide the numerator and denominator by the highest power of x in the denominator.

• Simplify each term and apply the property where n is a constant.

• Example: For f(x) = \frac{2x^2 + 3}{x^2 - 1}, as x \to \infty, .

Theorem: End Behavior and Asymptotes of Rational Functions

The end behavior of rational functions depends on the degrees of the numerator and denominator:

• If degree of numerator < degree of denominator: (horizontal asymptote at y = 0).

• If degree of numerator = degree of denominator: (horizontal asymptote at y = \frac{a_n}{b_n}).

• If degree of numerator > degree of denominator: is infinite (no horizontal asymptote).

Example: Identifying Asymptotes

For f(x) = \frac{2x^2 + 3}{x^2 - 1}, the horizontal asymptote is y = 2. For f(x) = \frac{x^3}{x^2 + 1}, there is no horizontal asymptote.

Summary Table: Horizontal Asymptotes of Rational Functions