The word "infinity" appears in two completely different limit contexts, and mixing them up causes consistent errors. The first: what happens to f(x) when x stays finite but f(x) grows without bound? The second: what does f(x) approach when x itself goes to infinity? Same notation family, completely different behaviour, different techniques.
Two Types — Don't Confuse Them
"Infinity" appears in limits in two distinct ways that students often conflate. They are fundamentally different concepts and require different techniques:
🔴 Infinite Limits
x approaches a finite value, but the function grows without bound. Written: lim (x→a) f(x) = ±∞. Describes a vertical asymptote at x = a.
🔵 Limits at Infinity
x approaches ±∞, and we ask what the function approaches. Written: lim (x→∞) f(x) = L. Describes a horizontal asymptote at y = L.
Infinite Limits (x → Finite, f(x) → ±∞)
An infinite limit occurs when the function blows up as x approaches a specific finite value. The most common cause is division by zero: as the denominator of a fraction approaches zero, the fraction's absolute value grows without bound.
limx→a⁺ 1/(x−a) = +∞
limx→a⁻ 1/(x−a) = −∞
To determine the sign of an infinite limit, substitute a value very slightly above or below a and check the sign of the fraction. The absolute value goes to infinity; the sign tells you which direction.
Analyse the infinite limits of f(x) = (x + 1) / (x − 3) near x = 3.
Limits at Infinity (x → ∞, f(x) → Finite)
A limit at infinity asks: what does f(x) settle toward as x grows without bound? If f(x) approaches a finite value L, then y = L is a horizontal asymptote. If f(x) → ±∞, there is no horizontal asymptote.
The key technique for limits at infinity of rational functions: divide numerator and denominator by the highest power of x in the denominator, then observe which terms vanish (those with x in the denominator → 0 as x → ∞).
Find lim (x→∞) [(3x² + 5x) / (2x² − 1)]
The Degree Rule for Rational Functions
For rational functions f(x) = p(x)/q(x) where p and q are polynomials:
| Condition | lim (x→∞) f(x) | Asymptote |
|---|---|---|
| deg(p) < deg(q) | 0 | y = 0 (x-axis) |
| deg(p) = deg(q) | leading coeff ratio | y = aₙ/bₙ |
| deg(p) > deg(q) | ±∞ | None (oblique if deg differs by 1) |
Growth Hierarchy — Which Function Wins at Infinity?
As x → ∞, different function types grow at vastly different rates. This hierarchy determines which term dominates in a limit:
- Logarithms (slowest): ln(x) grows incredibly slowly
- Polynomials: xⁿ grows faster as n increases
- Exponentials: eˣ dominates any polynomial — lim (x→∞) xⁿ/eˣ = 0 for any n
- Factorial / super-exponential: n! grows even faster than eⁿ
In a limit at infinity, the function with the fastest growth rate wins. Exponentials always beat polynomials. Polynomials always beat logarithms. This is why lim (x→∞) [xⁿ/eˣ] = 0 for any fixed n — exponentials eventually outrun any polynomial.
- Stewart, J. (2015). Calculus, §2.6. Cengage.
- Spivak, M. (2006). Calculus, Ch. 5–6. Publish or Perish.
- Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.