Limits at Infinity and Asymptotes: Study Notes

Limits at Infinity and Asymptotes

Introduction

In calculus, understanding the behavior of functions as the input variable approaches infinity or negative infinity is essential. This topic covers the definitions and properties of limits at infinity, infinite limits, and the concept of horizontal and slant asymptotes for polynomial and rational functions.

Limits at Infinity

Definition of Limits at Infinity

• Limit at Infinity: The notation $\lim_{x \to \infty} f(x) = L$ means that as $x$ increases without bound, $f(x)$ approaches the real number $L$.

• Limit at Negative Infinity: $\lim_{x \to -\infty} f(x) = M$ means that as $x$ decreases without bound, $f(x)$ approaches the real number $M$.

• Infinite Limit at Infinity: $\lim_{x \to \infty} f(x) = \infty$ means that $f(x)$ increases without bound as $x$ increases without bound.

• Infinite Limit at Negative Infinity: $\lim_{x \to -\infty} f(x) = \infty$ means that $f(x)$ increases without bound as $x$ decreases without bound.

• Negative Infinite Limit at Infinity: $\lim_{x \to \infty} f(x) = -\infty$ means that $f(x)$ decreases without bound as $x$ increases without bound.

• Negative Infinite Limit at Negative Infinity: $\lim_{x \to -\infty} f(x) = -\infty$ means that $f(x)$ decreases without bound as $x$ decreases without bound.

Limits at Infinity of Polynomials

For a polynomial $p(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$, where $a_n \neq 0$, the limit at infinity is determined by the leading term:

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

Note: When evaluating the infinite limit of a polynomial, focus on the leading term, as it dominates the behavior for large $|x|$.

Example 1: Find the following limits, if possible.

1. $\lim_{x \to \infty} 2x^4 - 5x + 6$ Solution: The leading term is $2x^4$, so the limit is $\infty$.

2. $\lim_{x \to -\infty} -3x^4$ Solution: The leading term is $-3x^4$. As $x \to -\infty$, $x^4$ is positive and large, so the limit is $-\infty$.

3. $\lim_{x \to \infty} \left(-5 - \frac{2}{x}\right)$ Solution: As $x \to \infty$, $\frac{2}{x} \to 0$, so the limit is $-5$.

Limits at Infinity of Rational Functions

For a rational function $r(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials, the behavior as $x \to \infty$ depends on the degrees of $p(x)$ and $q(x)$.

• Divide every term by the highest power of $x$ in the denominator to simplify the limit.

Example 2: Find the following limits, if possible.

1. $\lim_{x \to \infty} \frac{x^2 + 2}{x^2 - 1}$ Solution: Divide numerator and denominator by $x^2$: $\lim_{x \to \infty} \frac{1 + 2/x^2}{1 - 1/x^2} = \frac{1}{1} = 1$

2. $\lim_{x \to \infty} \frac{4x^2 + x + 6}{3x^2 + 7}$ Solution: Divide numerator and denominator by $x^2$: $\lim_{x \to \infty} \frac{4 + 1/x + 6/x^2}{3 + 7/x^2} = \frac{4}{3}$

3. $\lim_{x \to \infty} \frac{2^x - 2x}{2x + 1}$ Solution: The numerator grows much faster than the denominator, so the limit is $\infty$.

Horizontal Asymptotes

Definition of Horizontal Asymptote

• Horizontal Asymptote (HA): The line $y = b$ is a horizontal asymptote of the graph of $f(x)$ if either $\lim_{x \to \infty} f(x) = b$ or $\lim_{x \to -\infty} f(x) = b$.

Horizontal asymptotes describe the end behavior of a function as $x$ approaches $\infty$ or $-\infty$.

Rules for Finding Horizontal Asymptotes of Rational Functions

• Degree of numerator < degree of denominator: $\lim_{x \to \infty} f(x) = 0$; $y = 0$ is the horizontal asymptote.

• Degree of numerator = degree of denominator: $\lim_{x \to \infty} f(x) = \frac{a_n}{b_m}$, where $a_n$ and $b_m$ are the leading coefficients; $y = \frac{a_n}{b_m}$ is the horizontal asymptote.

• Degree of numerator > degree of denominator: $\lim_{x \to \infty} f(x) = \infty$ or $-\infty$; no horizontal asymptote.

Slant (Oblique) Asymptotes

• If the degree of the numerator is exactly one more than the degree of the denominator, the function has a slant asymptote of the form $y = mx + b$.

Example 3: Identify the horizontal asymptotes.

• $f(x) = \frac{1 - 2x^2}{2x^2 - x^2}$ Solution: Simplify denominator: $2x^2 - x^2 = x^2$. So $f(x) = \frac{1 - 2x^2}{x^2}$. As $x \to \infty$, the leading term is $-2x^2/x^2 = -2$. Thus, $y = -2$ is the horizontal asymptote.

• $f(x) = \frac{2x^2 + 5}{3x^2 + 1}$ Solution: Degrees are equal; leading coefficients are 2 and 3. So $y = \frac{2}{3}$ is the horizontal asymptote.

• $f(x) = \frac{50x^4 + 4x^2 - 1}{10x^4 + x^2 + 1}$ Solution: Degrees are equal; leading coefficients are 50 and 10. So $y = 5$ is the horizontal asymptote.

Vertical Asymptotes

Vertical asymptotes occur at values of $x$ where the denominator of a rational function is zero and the numerator is nonzero. For each vertical asymptote $x = a$, evaluate $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ to determine the behavior near the asymptote.

Summary Table: Asymptotes of Rational Functions

Degree of Numerator	Degree of Denominator	Horizontal Asymptote	Slant Asymptote
Less than	Greater than	$y = 0$	None
Equal	Equal	$y = \frac{a_n}{b_m}$	None
One more	Less by 1	None	$y = mx + b$
Greater by 2 or more	Less by 2 or more	None	None

Additional info: The above table summarizes the rules for determining horizontal and slant asymptotes for rational functions based on the degrees of the numerator and denominator.