Limit at Infinity

Tags: #◇Computer Science #◇CSMATH1 #◇Conceptual

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◇Overview

A limit at infinity describes the behavior of a function $f(x)$ as the independent variable $x$ increases without bound ($x \to +\infty$) or decreases without bound ($x \to -\infty$). This concept is useful for finding the long-term or end behavior of a function.

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

Limits of Rational Functions at Infinity:
To evaluate the limit of a rational function $f(x) = \frac{P(x)}{Q(x)}$ as $x \to \pm\infty$, divide every term in the numerator and denominator by the highest power of $x$ present in the denominator, or use the short-cut rules based on the degrees of $P(x)$ and $Q(x)$.

◇Important Formulas

Horizontal Asymptote Definition:
The line $y = L$ is a horizontal asymptote if:
$\lim_{x \to \infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L$
Reciprocal Rule:
For any rational number $r > 0$,
$\lim_{x \to \infty} \frac{1}{x^r} = 0$
If $x^r$ is defined for negative numbers, then:
$\lim_{x \to -\infty} \frac{1}{x^r} = 0$

◇Worked Examples

Example 1: Rational function where degrees are equal
Evaluate $\lim_{x \to \infty} \frac{3x^2 - 5x + 2}{4x^2 + 1}$.
Solution:
Divide the numerator and the denominator by $x^2$ (the highest power of $x$):
$\lim_{x \to \infty} \frac{3 - \frac{5}{x} + \frac{2}{x^2}}{4 + \frac{1}{x^2}}$
Using the property $\lim_{x \to \infty} \frac{1}{x^r} = 0$:
$= \frac{3 - 0 + 0}{4 + 0} = \frac{3}{4}$

Example 2: Rational function where denominator degree is larger
Evaluate $\lim_{x \to \infty} \frac{2x + 5}{x^2 - 3x + 1}$.
Solution:
Divide by $x^2$:
$\lim_{x \to \infty} \frac{\frac{2}{x} + \frac{5}{x^2}}{1 - \frac{3}{x} + \frac{1}{x^2}}$
$= \frac{0 + 0}{1 - 0 + 0} = 0$

Example 3: Rational function where numerator degree is larger
Evaluate $\lim_{x \to \infty} \frac{x^3 + 2}{x^2 + x}$.
Solution:
Divide by $x^2$:
$\lim_{x \to \infty} \frac{x + \frac{2}{x^2}}{1 + \frac{1}{x}}$
As $x \to \infty$, the numerator goes to $\infty$ while the denominator goes to $1$.
Thus, $\lim_{x \to \infty} \frac{x^3 + 2}{x^2 + x} = +\infty$.

◇References & Navigation

• Previous Topic: ◇One-Sided Limits and Infinite Limits
• Next Topic: ◇Continuity
• Related Notes: ◇Introduction to Limits, ◇One-Sided Limits and Infinite Limits