Limits at Infinity and Infinite Limits
A limit at infinity describes the behaviour of f(x) as x grows without bound (x → ∞ or x → −∞), and an infinite limit describes values of f that grow without bound as x approaches a finite point. In brief: limx→∞ (3x + 1)/(x − 2) = 3 (a horizontal asymptote y = 3), limx→2⁺ 1/(x − 2) = +∞ and limx→2⁻ 1/(x − 2) = −∞ (a vertical asymptote x = 2), and limx→∞ x2 = ∞.
These limits tell the long-term behaviour of a function and its behaviour near a singularity. They are the analytic basis for horizontal and vertical asymptotes. For rational functions, a short rule based on degrees gives the answer at once, and for other functions a comparison of growth rates helps. This guide gives the definitions, the rules for polynomials and rational functions, the limits of exponentials and logarithms, how to resolve ∞ − ∞ and ∞/∞ and a table of growth rates.
What is a limit at infinity?
limx→∞ f(x) = L means: for every ε > 0 there is M such that |f(x) − L| < ε for all x > M. The graph then approaches the horizontal line y = L. Similarly for x → −∞. The statement limx→∞ f(x) = ∞ means that f(x) exceeds any bound for large x.
The basic limits: limx→∞ 1/xp = 0 for p > 0; limx→∞ xp = ∞; limx→∞ ex = ∞ and limx→−∞ ex = 0; limx→∞ ln x = ∞; limx→∞ arctan x = π/2.
How do you find limits of rational functions at infinity?
For f(x) = P(x)/Q(x) with deg P = n and deg Q = m, divide numerator and denominator by xm, the highest power of the denominator.
- n < m: the limit is 0.
- n = m: the limit is the ratio of the leading coefficients.
- n > m: the limit is ±∞ (the sign depends on the leading coefficients and on the direction).
Example 1
(3x2 − x + 5)/(2x2 + 7) = (3 − 1/x + 5/x2)/(2 + 7/x2) → 3/2. And (x + 1)/(x3 − 4) → 0. And (2x3 + 1)/(x2 + 1) → +∞ as x → ∞ and −∞ as x → −∞.
Example 2: with roots
limx→∞ √(4x2 + 1)/(x + 3) = lim √(4 + 1/x2)/(1 + 3/x) = 2. As x → −∞, √(x2) = |x| = −x, so √(4x2 + 1)/(x + 3) = |x|√(4 + 1/x2)/(x(1 + 3/x)) = −√(4 + 1/x2)/(1 + 3/x) → −2. The two limits differ: the graph has two different horizontal asymptotes y = 2 and y = −2, which can occur for functions with roots, though not for rational functions.
What is an infinite limit?
limx→a f(x) = +∞ means: for every M there is δ > 0 such that f(x) > M whenever 0 < |x − a| < δ. The line x = a is then a vertical asymptote. The limit is called infinite, but it does not exist as a real number; the notation describes how the limit fails to exist.
Example 3
limx→0 1/x2 = +∞ (both sides positive). limx→0⁺ 1/x = +∞ and limx→0⁻ 1/x = −∞, so limx→0 1/x does not exist, though the one-sided limits are infinite.
Example 4: sign analysis
limx→3⁺ (x + 1)/(x − 3): the numerator tends to 4 > 0 and the denominator tends to 0 through positive values, so the limit is +∞. limx→3⁻: denominator negative, so −∞. For limx→1 (2 − x)/(x − 1)2: numerator → 1 > 0 and the denominator → 0⁺ from both sides, so the limit is +∞.
Example 5: ln near 0
limx→0⁺ ln x = −∞. And limx→(π/2)⁻ tan x = +∞.
How do functions compare in growth?
As x → ∞, for a > 1 and p > 0: ln x grows slower than xp, which grows slower than ax. In limit form: lim ln x/xp = 0 and lim xp/ax = 0. For polynomials, the term of highest degree dominates. These comparisons are proved with l’Hôpital’s rule.
Example 6
limx→∞ x5/ex = 0, since the exponential grows faster. And limx→∞ ln x/√x = 0. And limx→∞ (x2 + ex)/(ex − 3x) = lim (x2/ex + 1)/(1 − 3x/ex) = 1.
How do you resolve ∞ − ∞ and ∞/∞?
The forms ∞/∞ and ∞ − ∞ are indeterminate: the answer can be anything, and the expression must be transformed.
Example 7: ∞ − ∞ with roots
limx→∞ (√(x2 + 3x) − x) = lim 3x/(√(x2 + 3x) + x) = lim 3/(√(1 + 3/x) + 1) = 3/2.
Example 8: ∞ − ∞ with fractions
limx→∞ (x − √(x2 − 4x)) = lim 4x/(x + √(x2 − 4x)) = lim 4/(1 + √(1 − 4/x)) = 2.
Example 9: exponentials
limx→∞ (3x + 2x)/(3x − 2x) = lim (1 + (2/3)x)/(1 − (2/3)x) = 1.
Example 10: the number e
limx→∞ (1 + 2/x)x = e2, and limx→∞ ((x + 1)/(x − 1))x = lim (1 + 2/(x − 1))x = e2.
Table of limits at infinity
| Function | x → ∞ | x → −∞ |
|---|---|---|
| 1/x | 0 | 0 |
| x2 | ∞ | ∞ |
| x3 | ∞ | −∞ |
| ex | ∞ | 0 |
| e−x | 0 | ∞ |
| ln x | ∞ | not defined |
| arctan x | π/2 | −π/2 |
| (3x + 1)/(x − 2) | 3 | 3 |
| sin x | no limit | no limit |
Where do these ideas come from?
Asymptotes were studied by Apollonius for the hyperbola. The treatment of infinity in analysis developed in the seventeenth to nineteenth centuries: John Wallis introduced the symbol ∞ in 1655, and Cauchy and Weierstrass made limits at infinity and infinite limits precise using the ε–M and ε–δ language. The extended real line and the one-point compactification of the plane (the Riemann sphere) came later.
Where are they used?
Long-term behaviour is the central question in many models: the average cost of a product tends to the variable cost per unit; the concentration of a drug tends to a steady level; the temperature of a cooling body tends to the room temperature. The size of a program’s running time for large inputs is described by growth rates, as with O(n log n) versus O(n2). In physics, the potential energy near a singular point tends to infinity.
Which mistakes are common?
Treating ∞ as a number
∞ − ∞, ∞/∞ and 0 · ∞ are not defined. Transform the expression.
Forgetting the sign at x → −∞
With roots, √(x2) = |x|, which is −x for negative x.
Applying the degree rule to non-rational functions
For exponential and logarithmic functions, compare growth rates instead.
Ignoring one-sided behaviour at a vertical asymptote
The function can go to +∞ on one side and −∞ on the other. Analyse the sign on each side.