Limits at Infinity and Horizontal Asymptotes 2.5

Limits at Infinity

Limits at Infinity of Polynomials

When evaluating the limit of a polynomial function as $x$ approaches infinity or negative infinity, the term with the highest degree dominates the behavior of the function.

• General Form: $p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0$, where $a_n \neq 0$

• Limit as $x \to \infty$: $\lim_{x \to \infty} p(x) = \lim_{x \to \infty} a_n x^n$

• Limit as $x \to -\infty$: $\lim_{x \to -\infty} p(x) = \lim_{x \to -\infty} a_n x^n$

Example:

• $\lim_{x \to \infty} (5x^4 - 3x^3 + 2) = \lim_{x \to \infty} 5x^4 = +\infty$

• $\lim_{x \to -\infty} (-5x^4 - 3x^3 + 2) = \lim_{x \to -\infty} -5x^4 = -\infty$

Limits at Infinity of Rational Functions

For rational functions, the behavior as $x$ approaches infinity depends on the degrees of the numerator and denominator polynomials.

• Case 1: Degree of numerator < degree of denominator: Limit is 0.

• Case 2: Degree of numerator = degree of denominator: Limit is the ratio of leading coefficients.

• Case 3: Degree of numerator > degree of denominator: Limit is $\pm\infty$ (does not exist).

Examples:

• $\lim_{x \to \infty} \frac{3x^3 + 2}{x^5 - 1} = \lim_{x \to \infty} \frac{3x^3}{x^5} = \lim_{x \to \infty} \frac{3}{x^2} = 0$

• $\lim_{x \to \infty} \frac{50x^4 + 4x^2 - 1}{10x^4 + 8x^2 + 1} = \frac{50}{10} = 5$

• $\lim_{x \to \infty} \frac{x^2 - 2x}{2x + 4} = \lim_{x \to \infty} \frac{x^2}{2x} = \lim_{x \to \infty} \frac{x}{2} = +\infty$

• $\lim_{x \to -\infty} \frac{x^2 - 2x}{2x + 4} = \lim_{x \to -\infty} \frac{x}{2} = -\infty$

Limits at Infinity of Functions with Negative Powers

Functions with terms like $\frac{1}{x^n}$, where $n > 0$, approach zero as $x$ approaches infinity.

• Example: $f(x) = 2 + \frac{10}{x^3}$

• $\lim_{x \to \infty} f(x) = 2 + 0 = 2$

• $\lim_{x \to -\infty} f(x) = 2 + 0 = 2$

Horizontal Asymptotes (H.A.)

Definition and Identification

A horizontal asymptote of a function $f(x)$ is a horizontal line $y = L$ that the graph of $f(x)$ approaches as $x$ tends to $\infty$ or $-\infty$.

• General Form: $f(x) = \frac{p(x)}{q(x)}$

Rules for Finding Horizontal Asymptotes

• Case 1: Degree of $p(x)$ < degree of $q(x)$ Horizontal Asymptote: $y = 0$ Example: $f(x) = \frac{3x^3 + 2}{x^5 - 1}$ H.A.: $y = 0$

• Case 2: Degree of $p(x)$ = degree of $q(x)$ Horizontal Asymptote: $y = \frac{\text{leading coefficient of } p(x)}{\text{leading coefficient of } q(x)}$ Example: $f(x) = \frac{50x^4 + 4x^2 - 1}{10x^4 + 8x^2 + 1}$ H.A.: $y = \frac{50}{10} = 5$

• Case 3: Degree of $p(x)$ > degree of $q(x)$ Horizontal Asymptote: Does Not Exist (DNE) Example: $f(x) = \frac{x^2 - 2x}{2x + 4}$ H.A.: DNE

Example: Finding Vertical and Horizontal Asymptotes

Consider $f(x) = \frac{x^2 - 81}{x(x - 9)}$

• Find the Vertical Asymptote (V.A.): Factor numerator: $x^2 - 81 = (x + 9)(x - 9)$ $f(x) = \frac{(x + 9)(x - 9)}{x(x - 9)} = \frac{x + 9}{x}$ (for $x \neq 9$) Vertical asymptote at $x = 0$

• Identify the Horizontal Asymptote (H.A.): $f(x) = \frac{x^2 - 81}{x^2 - 9x}$ Leading coefficients: $1$(numerator), $1$ (denominator) $\text{H.A.}: y = \frac{1}{1} = 1$

Summary Table: Horizontal Asymptotes of Rational Functions

Degree of Numerator	Degree of Denominator	Horizontal Asymptote	Example
< Denominator	Higher	$y = 0$	$\frac{3x^3 + 2}{x^5 - 1}$
= Denominator	Equal	$y = \frac{\text{leading coeff. numerator}}{\text{leading coeff. denominator}}$	$\frac{50x^4 + 4x^2 - 1}{10x^4 + 8x^2 + 1}$
> Denominator	Lower	DNE	$\frac{x^2 - 2x}{2x + 4}$

Key Terms

• Limit at Infinity: The value a function approaches as $x$ increases or decreases without bound.

• Horizontal Asymptote: A horizontal line that the graph of a function approaches as $x \to \infty$ or $x \to -\infty$.

• Vertical Asymptote: A vertical line $x = a$ where the function grows without bound as $x$ approaches $a$.

Additional info:

• All examples and explanations are based on standard calculus concepts for limits at infinity and asymptotes.

• Graphs referenced in the notes illustrate the behavior of rational functions near their asymptotes.