Mathematical formulas

Differential Calculus

Subjects Home Search Guide

Sections

1Notation, functions and domains
2Limits
3Continuity
4Derivative and geometric interpretation
5Differentiation rules
6Higher-order derivatives
7Mean value theorem
8Function analysis
9Optimization
10Linear approximations and differentials
11Taylor series (introduction)
12L'Hôpital and indeterminate limits
13Implicit and related functions
14Related rates
15Graphs and asymptotic behavior
16Guide to differentiate and analyze
AAppendix A: extended derivative table

Search formulas…

Language

Loading formula…

Mathematical formulas · API-served content · Guide

About Reviews Contact

Home
Differential Calculus
Limits at infinity
Formula

2.6. Limits at infinity

Formula

In 2.6. Limits at infinity

Formula

\lim_{x\to\infty}\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots+b_0}= \begin{cases} 0, & n<m\\ \frac{a_n}{b_m}, & n=m\\ \operatorname{sgn}\!\left(\frac{a_n}{b_m}\right)\cdot\infty, & n>m \end{cases}

Tags

Differential calculus
Limit
Limits at infinity

Detail

Behavior of polynomial quotients as

x\to\infty

(same for

x\to-\infty

when degrees differ; if

n>m

, the sign depends on

a_n/b_m

and the direction).

Variables

$x$ Independent variable
$a$ Point or real constant
$b$ Interval endpoint or constant
$n$ Derivative order or exponent

Found an error? Contact us.

Related formulas

Related
Formula
$\lim_{x\to\infty}\frac{1}{x^p}=0,\qquad p>0$
Open formula Open in section