Mathematics.

limits

Limits at Infinity and Horizontal Asymptotes

Calculus I35 minDifficulty4 out of 10

You should know: limit, functions

Overview

Limits at infinity describe the long-run behavior of a function: lim_{x->inf} f(x) = L means f(x) approaches L as x grows without bound. When such a limit L exists, the line y=L is a horizontal asymptote of the graph. Key facts: rational functions p(x)/q(x) have limit equal to the ratio of leading coefficients when deg(p)=deg(q), 0 when deg(p)<deg(q), and +/-infinity when deg(p)>deg(q). Exponentials beat polynomials: e^x -> inf faster than any x^n.

Intuition

Limits at infinity ask: where does the function 'settle' as you zoom far to the right (or left)? For rational functions, only the highest-degree terms matter at large x -- lower-order terms become negligible. For 1/x: as x gets huge, 1/x approaches 0 from above. For (2x+1)/(x-3): divide top and bottom by x to get (2+1/x)/(1-3/x) -> 2/1 = 2 as x -> infinity.

Formal Definition

Definition

lim_{x->inf} f(x) = L means: for every epsilon > 0 there exists N > 0 such that if x > N then |f(x) - L| < epsilon. Similarly, lim_{x->-inf} f(x) = L uses: for every epsilon > 0 there exists N < 0 such that x < N implies |f(x)-L| < epsilon. lim_{x->inf} f(x) = inf means: for every M > 0 there exists N > 0 such that x > N implies f(x) > M.

lim⁡x→∞f(x)=L  ⟺  ∀ε>0  ∃N:  x>N⇒∣f(x)−L∣<ε\lim_{x\to\infty} f(x) = L \iff \forall\varepsilon>0\;\exists N:\; x>N \Rightarrow |f(x)-L|<\varepsilon
Formal definition
lim⁡x→∞anxn+⋯bmxm+⋯={an/bnn=m0n<m±∞n>m\lim_{x\to\infty}\frac{a_n x^n + \cdots}{b_m x^m + \cdots} = \begin{cases} a_n/b_n & n=m \\ 0 & n<m \\ \pm\infty & n>m \end{cases}
Rational function limits
lim⁡x→∞p(x)ex=0 for any polynomial p(x)\lim_{x\to\infty}\frac{p(x)}{e^x} = 0 \text{ for any polynomial } p(x)
Exponentials dominate polynomials

Notation

NotationMeaning
lim⁡x→∞\lim_{x\to\infty}Limit as x grows without bound
lim⁡x→−∞\lim_{x\to-\infty}Limit as x decreases without bound
y=Ly = LHorizontal asymptote

Theorems

Theorem 1: Limits of Basic Functions at Infinity
limx−>inf1/xp=0foranyp>0.limx−>infex=inf.limx−>−infex=0.limx−>infln(x)=inf(butslowerthananyxepsilon).limx−>infarctan(x)=pi/2.limx−>−infarctan(x)=−pi/2.lim_{x->inf} 1/x^p = 0 for any p > 0. lim_{x->inf} e^x = inf. lim_{x->-inf} e^x = 0. lim_{x->inf} ln(x) = inf (but slower than any x^epsilon). lim_{x->inf} arctan(x) = pi/2. lim_{x->-inf} arctan(x) = -pi/2.
Theorem 2: Limit at Infinity for Rational Functions
Letf(x)=(anxn+lower)/(bmxm+lower).Then:ifn=m,limx−>inff(x)=an/bm;ifn<m,limx−>inff(x)=0;ifn>m,limx−>inff(x)=+/−infinity(signdependsonleadingcoefficientsignsandparityofn−m).Let f(x) = (a_n x^n + lower)/(b_m x^m + lower). Then: if n=m, lim_{x->inf} f(x) = a_n/b_m; if n<m, lim_{x->inf} f(x) = 0; if n>m, lim_{x->inf} f(x) = +/-infinity (sign depends on leading coefficient signs and parity of n-m).
Theorem 3: Growth Hierarchy
Asx−>inf,thefollowinghold:ln(x)<<xalpha(anyalpha>0)<<ax(anya>1)<<xx.Moreprecisely,foranyalpha>0anda>1:limx−>infln(x)/xalpha=0;limx−>infxn/ex=0foralln;limx−>infax/bx=0whenevera<b.As x -> inf, the following hold: ln(x) << x^alpha (any alpha>0) << a^x (any a>1) << x^x. More precisely, for any alpha > 0 and a > 1: lim_{x->inf} ln(x)/x^alpha = 0; lim_{x->inf} x^n/e^x = 0 for all n; lim_{x->inf} a^x/b^x = 0 whenever a < b.

Worked Examples

  1. 1

    Divide numerator and denominator by x^2 (the highest power in the denominator).

    3x2−2x+15x2+4=3−2/x+1/x25+4/x2\frac{3x^2 - 2x + 1}{5x^2 + 4} = \frac{3 - 2/x + 1/x^2}{5 + 4/x^2}
  2. 2

    As x->inf: 2/x -> 0, 1/x^2 -> 0, 4/x^2 -> 0.

  3. 3

    So the limit = (3 - 0 + 0)/(5 + 0) = 3/5.

    lim⁡x→∞3x2−2x+15x2+4=35\lim_{x\to\infty} \frac{3x^2 - 2x + 1}{5x^2 + 4} = \frac{3}{5}

✓ Answer

3/5. The line y = 3/5 is a horizontal asymptote.

Practice Problems

Easyapplication

Find all horizontal asymptotes of f(x) = (x^2 - 1)/(x^2 + 1).

Common Mistakes

Common Mistake

Thinking lim_{x->inf} sin(x) = 0 because sin oscillates 'smaller'.

sin(x) oscillates between -1 and 1 forever; it does NOT converge to 0. lim_{x->inf} sin(x) does not exist. However, lim_{x->inf} sin(x)/x = 0 (squeeze: -1/x <= sin(x)/x <= 1/x, and both bounds -> 0). The function itself vs. the function divided by x behave differently.

Quiz

What is lim_{x->inf} (2x^3 - 5) / (x^3 + 7x)?

Historical Background

The concept of limits at infinity was implicit in Leibniz's and Newton's use of infinitely large quantities. Cauchy formalized limits in the finite case; the extension to infinity followed naturally. The notion of asymptotes dates to ancient Greek geometry (curves approaching lines). Modern calculus textbooks treat horizontal asymptotes via limits at infinity as a standard topic in the study of curve sketching and function behavior.

  1. 1821

    Cauchy formalizes limit theory including limits at infinity

    Augustin-Louis Cauchy

  2. 1860s

    Weierstrass provides rigorous epsilon-N definition of limits at infinity

    Karl Weierstrass

Summary

  • lim_{x->inf} f(x) = L means f(x) eventually stays within epsilon of L for large enough x.
  • Rational functions: ratio of leading terms determines limit at infinity.
  • Growth hierarchy: ln(x) << polynomials << exponentials.
  • Horizontal asymptotes y=L occur when lim_{x->+/-inf} f(x) = L.

References

  1. BookStewart, J. Calculus: Early Transcendentals. 8th ed. Cengage, 2015.
  2. WebsiteWikipedia -- Asymptote