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Calculus I / Limits and Continuity

Limits at Infinity, Horizontal Asymptotes, and Asymptotic Comparison

What happens to a bond discount factor e−rTe^{-rT} as maturity T→∞T \to \infty? What is the long-run behaviour of an implied volatility surface that is quoted in terms of log-moneyness? What does a call price approach as the strike goes to zero? All of these questions are answered by limits at infinity — the behaviour of a function as its argument grows without bound. This note defines lim⁡x→±∞f(x)\lim_{x\to\pm\infty} f(x), introduces horizontal asymptotes, establishes the key technique for rational functions (divide by the highest power), and closes with the growth-rate hierarchy that governs how different function families compare at ±∞\pm\infty.
Learning objectives. By the end of this note you should be able to:
  • state and apply the definitions of lim⁡x→+∞f(x)=L\lim_{x\to+\infty} f(x) = L and lim⁡x→−∞f(x)=L\lim_{x\to-\infty} f(x) = L;
  • identify horizontal asymptotes from a limit computation;
  • evaluate limits at infinity for rational functions by dividing numerator and denominator by the highest power;
  • state and apply the growth-rate hierarchy ln⁡x≪xα≪ex\ln x \ll x^\alpha \ll e^x as x→∞x\to\infty;
  • evaluate limits involving exe^x and ln⁡x\ln x at ±∞\pm\infty.

In this lesson

  1. Limits at infinity
  2. Rational functions at infinity
  3. Limits of exe^x and ln⁡x\ln x at ±∞\pm\infty
  4. The growth-rate hierarchy
  5. Limits at −∞-\infty and oscillating functions

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