Limits at Infinity and Asymptotes: A Calculus Study Guide

Limits at Infinity and Asymptotes

Introduction

This study guide covers the concepts of limits at infinity, infinite limits, and asymptotes, which are fundamental in understanding the end behavior of functions in calculus. These topics are essential for analyzing graphs, determining asymptotic behavior, and evaluating limits involving rational and trigonometric functions.

Limits at Infinity

Definition and Notation

• Limit at Infinity describes the value that a function approaches as the independent variable (usually x) increases or decreases without bound.

• Notation: $\lim_{x \to \infty} f(x) = L$ means as $x$ becomes arbitrarily large, $f(x)$ approaches $L$.

• Similarly, $\lim_{x \to -\infty} f(x) = L$ for $x$ becoming arbitrarily negative.

Horizontal Asymptotes

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

• If $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$, then $y = L$ is a horizontal asymptote.

Examples

• For $f(x) = 2 + \frac{1}{x}$:

  • $\lim_{x \to \infty} f(x) = 2$

  • $\lim_{x \to -\infty} f(x) = 2$

  • Thus, $y = 2$ is a horizontal asymptote.

Infinite Limits and Vertical Asymptotes

Definition and Notation

• Infinite Limit occurs when $f(x)$ increases or decreases without bound as $x$ approaches a finite value $a$.

• Notation: $\lim_{x \to a} f(x) = \infty$ or $-\infty$.

• One-sided limits: $\lim_{x \to a^+} f(x)$ (from the right), $\lim_{x \to a^-} f(x)$ (from the left).

Vertical Asymptotes

• A vertical asymptote is a vertical line $x = a$ where $f(x)$ increases or decreases without bound as $x$ approaches $a$.

• If $\lim_{x \to a^+} f(x) = \pm\infty$ or $\lim_{x \to a^-} f(x) = \pm\infty$, then $x = a$ is a vertical asymptote.

Examples

• For $f(x) = \frac{1}{x^2}$:

  • $\lim_{x \to 0^+} f(x) = \infty$

  • $\lim_{x \to 0^-} f(x) = \infty$

  • Thus, $x = 0$ is a vertical asymptote.

Comparing Infinite Limits and Limits at Infinity

Infinite Limit	Limit at Infinity
Dependent variable $f(x)$ approaches infinity as independent variable $x$ approaches a finite number.	Independent variable $x$ approaches infinity; dependent variable $f(x)$ approaches a finite number.
$\lim_{x \to a} f(x) = \infty$	$\lim_{x \to \infty} f(x) = a$

End Behavior of Polynomials

Even and Odd Degree Polynomials

• For $f(x) = x^n$:

  • If $n$ is even:

    • $\lim_{x \to \infty} x^n = \infty$

    • $\lim_{x \to -\infty} x^n = \infty$

  • If $n$ is odd:

    • $\lim_{x \to \infty} x^n = \infty$

    • $\lim_{x \to -\infty} x^n = -\infty$

Limits of Rational Functions at Infinity

General Form

For a rational function $f(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials:

• If degree of numerator < degree of denominator: $\lim_{x \to \pm\infty} f(x) = 0$

• If degree of numerator = degree of denominator: $\lim_{x \to \pm\infty} f(x) = \frac{a_n}{b_m}$, where $a_n$ and $b_m$ are leading coefficients.

• If degree of numerator > degree of denominator: $\lim_{x \to \pm\infty} f(x) = \pm\infty$ (sign depends on leading coefficients and direction).

Examples

• $\lim_{x \to \infty} \frac{2x^3}{x^3} = 2$

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

• $\lim_{x \to \infty} \frac{1 - 3x^4 + 2x^9}{4x^9 - x^5 - 5x^6} = \frac{2}{4} = \frac{1}{2}$

Finding Vertical Asymptotes

• Vertical asymptotes occur at values of $x$ where the denominator of a rational function is zero and the numerator is nonzero.

• For $f(x) = \frac{x^2 - 9x + 14}{x^2 - 5x + 6}$, set denominator $x^2 - 5x + 6 = 0$ to find $x = 2, 3$.

Practice Problems and Applications

• Evaluate $\lim_{x \to 5^+} \frac{3}{(x-5)^4}$: As $x \to 5^+$, denominator approaches $0^+$, so the limit is $+\infty$.

• Evaluate $\lim_{x \to 1^-} \frac{x}{(x^2 - 1)^2}$: As $x \to 1^-$, denominator approaches $0^+$, numerator approaches $1$, so the limit is$+\infty$.

• Evaluate $\lim_{x \to 1^+} \frac{x}{(x^2 - 1)^2}$: As $x \to 1^+$, denominator approaches $0^+$, numerator approaches $1$, so the limit is$+\infty$.

• Evaluate $\lim_{x \to 1} \frac{x}{(x^2 - 1)^2}$: Since both one-sided limits are $+\infty$, the two-sided limit is $+\infty$.

Special Trigonometric Limits

• Evaluate $\lim_{\theta \to 0} \frac{\sin^2 \theta}{1 - \cos^2 \theta}$:

  • $1 - \cos^2 \theta = \sin^2 \theta$

  • So, $\frac{\sin^2 \theta}{\sin^2 \theta} = 1$ for $\theta \neq 0$

  • Thus, the limit is $1$.

Summary Table: Asymptotes of Rational Functions

Degree of Numerator	Degree of Denominator	Horizontal Asymptote	Oblique Asymptote	Vertical Asymptote
Less than	Greater than	$y = 0$	None	Set denominator = 0
Equal	Equal	$y = \frac{a_n}{b_m}$	None	Set denominator = 0
Greater than	Less than	None	Possible (if degree is one more)	Set denominator = 0

Key Terms

• Limit at Infinity: The value a function approaches as $x$ becomes very large or very small.

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

• Horizontal Asymptote: A horizontal line that the graph approaches as $x \to \pm\infty$.

• Vertical Asymptote: A vertical line where the function increases or decreases without bound.

Additional info:

• Oblique (slant) asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator.

• For rational functions, always reduce to lowest terms before determining asymptotes.