Limits at Infinity

Definition

Limits at infinity describe the “end behavior” of a function $f(x)$ as the input $x$ increases without bound ($x \to \infty$) or decreases without bound ($x \to -\infty$).

Why It Matters

Every system has an “end behavior.” Limits at infinity allow us to predict the horizontal asymptotes of reality—how a population settles, how a heat wave dissipates, or how a business model scales when pushed to its ultimate boundary.

Core Concepts

• Convergent Behavior: $\lim_{x \to \pm \infty} f(x) = L$ indicates the function approaches a horizontal line $y = L$.

  • How to read: “The limit as x approaches plus or minus infinity of f of x equals L.”
  • Meaning: End behavior settles at height $L$—defines horizontal asymptote $y = L$.

• Basic Rules:

  • $\lim_{x \to \pm \infty} k = k$ (for any constant $k$)
    • How to read: “The limit of a constant k as x approaches infinity equals k.”
    • Meaning: A flat function stays flat forever.
  • $\lim_{x \to \pm \infty} \frac{1}{x} = 0$
    • How to read: “The limit of one over x as x approaches infinity equals zero.”
    • Meaning: Reciprocals of growing inputs vanish—foundation for rational function end behavior.

• Rational Functions: The limit is determined by the ratio of the highest-degree terms in the numerator and denominator.

Connected Concepts

• Fundamental Trigonometric Limits
• Continuity at a Point
• Derivative at a Point