2.6 Limits at Infinity and Horizontal Asymptotes - Quick Notes

Core Concepts

  • End behavior: lim_{x→±∞} f(x) describes how f behaves as x becomes very large in magnitude.
  • Horizontal asymptote: a horizontal line y = L that the graph approaches as x→±∞, i.e. lim_{x→±∞} f(x) = L.
  • Notation for infinity limits: \lim{x→∞} f(x) = L\;\text{and/or}\;\lim{x→-∞} f(x) = L.
  • Infinite limits at infinity: limits that grow without bound, e.g. lim_{x→∞} f(x) = ∞ or = -∞.
  • Key example structure: evaluate end behavior of common functions to identify horizontal asymptotes.

Horizontal Asymptotes: Definitions and Notation

  • If either \lim{x→∞} f(x) = L or \lim{x→-∞} f(x) = L holds, then the line y = L is a horizontal asymptote.
  • Example: for f(x) = (x^2 - 1)/(x^2 + 1),
    \lim_{x→∞} f(x) = 1.
    Hence y = 1 is a horizontal asymptote.
  • Example: \lim{x→∞} \frac{1}{x} = 0 \quad \text{and} \quad \lim{x→-∞} \frac{1}{x} = 0, so y = 0 is a horizontal asymptote.
  • Example: For f(x) = 3x, \lim{x→∞} f(x) = ∞\quad\text{and}\quad \lim{x→-∞} f(x) = -∞.

How to Find Horizontal Asymptotes of Rational Functions

  • For a rational function f(x) = P(x)/Q(x):
    • If deg(P) < deg(Q): horizontal asymptote at y = 0.
    • If deg(P) = deg(Q): horizontal asymptote at y = (leading coefficient of P)/(leading coefficient of Q).
    • If deg(P) > deg(Q): typically no horizontal asymptote (may have an oblique asymptote).
  • Method (end behavior for ∞): divide numerator and denominator by the highest power of x in the denominator; terms with x in the denominator vanish as x→∞.

Example Illustrations

  • f(x) = (x^2 - 1)/(x^2 + 1) ⇒ \lim_{x→∞} f(x) = 1 (horizontal asymptote y = 1).
  • f(x) = 1/x ⇒ \lim{x→∞} f(x) = 0, \; \lim{x→-∞} f(x) = 0. (horizontal asymptote y = 0)
  • For the function f(x) = 2x^3 + … over deg 2 denominator (Example 4 in notes), the horizontal asymptote is found by comparing leading terms after division; often results in a finite y = L.

Precise Definitions (ε–N)

  • Definition (limit at ∞): Let f be defined on (a, ∞). Then
    \lim_{x→∞} f(x) = L
    iff for every \varepsilon > 0 there exists N such that if x > N then |f(x) - L| < \varepsilon.
  • Definition (limit at -∞): Let f be defined on (−∞, a). Then
    \lim_{x→-∞} f(x) = L
    iff for every \varepsilon > 0 there exists N such that if x < N then |f(x) - L| < \varepsilon.
  • Summary: y = L is a horizontal asymptote if either of the above limits equals L.

Infinite Limits at Infinity (Rigorous View)

  • Notation:
    • \lim{x→∞} f(x) = ∞, \lim{x→∞} f(x) = -∞
    • \lim{x→-∞} f(x) = ∞, \lim{x→-∞} f(x) = -∞
  • Definitions (examples):
    • If \lim_{x→∞} f(x) = ∞, then for every M > 0 there exists N such that x > N ⇒ f(x) > M.
    • If \lim_{x→∞} f(x) = -∞, then for every M > 0 there exists N such that x > N ⇒ f(x) < -M.
    • If \lim_{x→-∞} f(x) = ∞, then for every M > 0 there exists N such that x < N ⇒ f(x) > M.
    • If \lim_{x→-∞} f(x) = -∞, then for every M > 0 there exists N such that x < N ⇒ f(x) < -M.
  • Example: For f(x) = 3x, as x → ∞, f(x) → ∞; as x → -∞, f(x) → -∞.

Example: Proving a Basic Limit with ε–N (1/x → 0)

  • Claim: \lim_{x→∞} \frac{1}{x} = 0.
  • Proof sketch: Given \varepsilon > 0, choose N with N > 1/\varepsilon. If x > N then
    |\frac{1}{x} - 0| = \frac{1}{x} < \frac{1}{N} < \varepsilon.
  • Hence the limit is 0.

Practical Takeaways

  • Horizontal asymptotes reflect end behavior of rational functions via leading terms.
  • To test end behavior, divide by the highest power of x in the denominator; assess deg(P) vs deg(Q).
  • Precise ε–N definitions provide the rigorous foundation for limits at ±∞.
  • Infinite limits are described by a similar ε–N style, but with unbounded target values (∞ or −∞).