§ 01What Does x → ∞ Actually Mean?
Infinity is not a number you can substitute into a function. It is a direction, a process of growing without any bound. Limits at infinity describe the destination of a function's output as its input takes that journey.
When we write limx→∞ f(x) = L, we are making a precise claim: for every positive tolerance ε, no matter how small, there exists a sufficiently large threshold M such that once x surpasses M, the value f(x) stays within ε of L forever. The function does not have to reach L: it has to approach it arbitrarily closely and stay close.
Similarly, limx→−∞ f(x) = L describes what happens as x decreases without bound: as x sprints off to the left along the number line without any floor to stop it.
This makes geometric sense: as x grows enormous, dividing 1 by a large number raised to a positive power produces something increasingly tiny. The reciprocal of a billion is essentially zero to any practical calculation, and the limit formalises this intuition exactly.
Horizontal Asymptotes — The Geometric Picture
When limx→∞ f(x) = L (a finite number), the line y = L is called a horizontal asymptote of f as x → +∞. Equivalently, if limx→−∞ f(x) = M, then y = M is a horizontal asymptote as x → −∞. A function can have at most two distinct horizontal asymptotes, one for each infinite direction, and it is entirely possible for both limits to equal the same value L (one asymptote) or for one or both limits to be infinite (no asymptote in that direction).
The graph hugs the line y = L from the right side of the picture.
The graph hugs the line y = M from the left side of the picture. L and M may differ.
A function with two different horizontal asymptotes, y = M as x → −∞, and y = L as x → +∞
§ 02Rational Functions — The Divide by Highest Power Technique
Rational functions (polynomial over polynomial) are the most common setting for limits at infinity, and they all follow the same three-case rule that flows directly from comparing degrees.
The key technique is to divide every single term in both numerator and denominator by the highest power of x that appears anywhere in the denominator. After dividing, every term of the form c/xⁿ (with n ≥ 1) vanishes to zero as x → ±∞, leaving only the surviving constants.
degree(top) = degree(bottom) → limit = ratio of leading coefficients
degree(top) > degree(bottom) → limit = ±∞ (no horizontal asymptote)
Memorising these three cases is useful shorthand, but understanding why they work, by doing the division, builds the instincts you need for more complex functions later.
Both numerator and denominator are degree 2. Divide every term by x² (highest power in denominator).
The limit equals the ratio of the leading coefficients, exactly as the rule predicts.
Limit = 3/5 (horizontal asymptote y = 3/5)Degree of numerator (1) is less than degree of denominator (2). Divide by x².
Degree of numerator (3) exceeds degree of denominator (2). Divide by x².
No horizontal asymptote in this direction. The function grows without bound.
Limit = ∞ (no horizontal asymptote)Same degree. The sign of x doesn't matter for terms that end up as constants after dividing.
§ 03Radical Functions — The Sign Trap
When square roots appear, the divide-by-highest-power technique still works, but you must be careful about the sign of √(x²), which depends on the direction x is travelling.
The crucial fact: for any real x, √(x²) = |x|. When x → +∞, x is positive so |x| = x. When x → −∞, x is negative so |x| = −x. Forgetting this sign flip when x → −∞ is the single most common error students make with radical limits.
The highest power in the denominator is x (since √(x²) = x for large positive x). Divide numerator and denominator by x.
Same function, opposite direction. Now x is large and negative, so √(x²) = −x.
This function has two different horizontal asymptotes: y = 3 as x → +∞ and y = −3 as x → −∞. The sign trap is why they differ.
Limit = −3 (note the sign flip!)§ 04Exponential and Logarithmic Limits at Infinity
Exponential functions either explode to infinity or collapse to zero: never anything in between. Knowing which direction depends entirely on the sign of the exponent.
limx→+∞ e−x = 0 · limx→−∞ e−x = +∞
limx→+∞ ln x = +∞ · limx→0⁺ ln x = −∞
When exponentials and polynomials compete in a limit, the exponential always dominates. No polynomial, however large its degree, can outrun eˣ as x → ∞. This is the heart of the dominance hierarchy:
| Rank | Function type | Example | Beats… |
|---|---|---|---|
| Slowest | Logarithms | ln x, log₁₀ x | Nothing grows slower (among common types) |
| ↑ | Powers / polynomials | x, x², x¹⁰⁰ | Dominates logarithms |
| ↑ | Exponentials | eˣ, 2ˣ, 10ˣ | Dominates all polynomials |
| Fastest | Factorials / super-exp | n!, xˣ | Dominates exponentials |
Polynomial x⁵ vs exponential eˣ. By dominance, eˣ wins: the denominator grows far faster than the numerator.
Logarithm vs polynomial (degree 1). The polynomial dominates. Both → ∞, so apply L'Hôpital once.
§ 05Trigonometric Functions at Infinity
Sine and cosine oscillate perpetually between −1 and 1. They never settle, so limx→∞ sin x does not exist, but that doesn't stop them appearing inside more complex limits.
When a bounded oscillating function (like sin x or cos x) is divided by a function that grows without bound, the Squeeze Theorem resolves the limit cleanly. The bounded numerator gets squeezed to zero by the growing denominator.
sin x oscillates between −1 and 1, while x → ∞. Apply the Squeeze Theorem.
The cos x term is bounded while x² grows. It will contribute nothing to the limit.
§ 06Common Mistakes
§ 07Practice Quiz — 10 Questions
Test your grasp of limits at infinity, rational functions, exponentials, radicals, and oscillating functions.
limx→∞ 1/x³ = ?
Answer:
limx→∞ (5x² − 3) / (2x² + 7x) = ?
limx→∞ (x + 1) / (x² − 4) = ?
What is limx→+∞ eˣ?
limx→+∞ (2x − 5) / √(4x² + 1) = ?
Answer:
limx→−∞ (2x − 5) / √(4x² + 1) = ?
Which function grows fastest as x → ∞?
limx→∞ cos x / x = ?
limx→∞ (7x³ − 2x + 1) / (3x³ + 5) = ?
Answer (as a fraction, enter decimal):
A function has limx→∞ f(x) = 4. What does this mean geometrically?
Next StepsContinue the Limits Series
Limits at infinity underpin horizontal asymptotes in curve sketching and appear throughout integration and series. Continue with:
- Indeterminate Forms, when both numerator and denominator grow together in an ambiguous competition.
- Continuity, how limits at ordinary points define the seamless behaviour of functions.
- Curve Sketching, where limits at infinity tell you the right-hand and left-hand tails of every graph.