1. What Limits at Infinity Mean
A limit at infinity describes how a function behaves as x becomes extremely large (towards +∞) or extremely small (towards −∞). Instead of approaching a point, we look at the long-term behaviour of the function.
This helps in understanding end behaviour, horizontal asymptotes, and growth rates of functions.
2. Two Types of Limits at Infinity
There are two directions in which x can grow:
2.1. As x → +∞
x becomes larger and larger. We study what value f(x) gets close to as x increases without bound.
2.2. As x → −∞
x becomes more and more negative. We watch how f(x) behaves when x decreases without bound.
3. Notation for Limits at Infinity
The standard notation is:
3.1. Notation
\lim_{x \to \infty} f(x)
\lim_{x \to -\infty} f(x)
4. Examples with Polynomials
Most polynomial behaviour at infinity is controlled by the highest power of x.
4.1. Example 1
f(x) = 3x^2 + 5x - 7
As x → +∞, the x² term dominates, so f(x) → +∞.
4.2. Example 2
f(x) = -2x^3 + x
As x → +∞, the −2x³ term dominates → f(x) → −∞.
5. Rational Functions and Horizontal Asymptotes
Limits at infinity are very useful for rational functions. Depending on degrees of numerator and denominator, the function may approach a constant value.
5.1. Case 1 — Degree (numerator) < Degree (denominator)
The limit is zero.
\lim_{x \to \infty} \frac{1}{x^2} = 0
5.2. Case 2 — Degrees Are Equal
The limit is the ratio of leading coefficients.
\lim_{x \to \infty} \frac{3x^2 + 1}{6x^2 - 5} = \frac{3}{6} = \frac{1}{2}
5.3. Case 3 — Degree (numerator) > Degree (denominator)
The function grows without bound (→ +∞ or → −∞ depending on signs).
6. Limits at Infinity for Roots and Rational Powers
Functions like square roots grow more slowly than polynomials.
6.1. Example
\lim_{x \to \infty} \sqrt{x} = \infty
7. Exponential vs Polynomial Growth
Exponential functions grow faster than any polynomial as x → +∞.
7.1. Example
\lim_{x \to \infty} \frac{e^x}{x^3} = \infty
8. Horizontal Asymptotes
If a function approaches a finite value as x → ±∞, that value represents a horizontal asymptote of the graph.
8.1. Example
\lim_{x \to \infty} \frac{4x+1}{2x+5} = 2
So y = 2 is a horizontal asymptote.
9. Why Limits at Infinity Matter
They help describe long-term behaviour, growth rates, and asymptotes. They are used throughout calculus to understand functions globally, not just near specific points.