Limit at Infinity Calculator

The degree-comparison rule for rational functions at infinity

For a rational function (one polynomial divided by another), the behavior as x → ±∞ depends entirely on comparing the degree of the numerator to the degree of the denominator; every lower-degree term becomes irrelevant once x is large enough, because the highest-degree term dominates the sum. This one comparison covers the overwhelming majority of "limit at infinity" problems in a typical calculus course:

• Numerator degree less than denominator degree: the limit is 0. Example: (2x + 1) / (x² + 3) → 0 as x → ∞, since the denominator grows faster.
• Numerator degree = denominator degree: the limit is the ratio of the leading coefficients. Example: (3x² + 1) / (x² - 5) → 3/1 = 3 as x → ∞.
• Numerator degree greater than denominator degree: the limit is +∞ or -∞ (a signed infinite limit, not a finite number). Example: (x³ + 1) / (x + 1) → +∞ as x → ∞.

Worked example, full walkthrough

Consider limx→∞ (3x² + 1) / (x² - 5). Both the numerator (degree 2) and the denominator (degree 2) have the same degree, so the equal-degree rule applies directly: divide the leading coefficients, 3 and 1, to get 3. A quick sanity check with large values confirms it: at x = 1,000,000 the ratio is extremely close to 3, and it keeps getting closer as x grows further, exactly matching the exact algebraic answer with no need to plug in an actual infinite value.

1. Identify the degrees: numerator 3x² + 1 has degree 2; denominator x² - 5 has degree 2.
2. Equal degrees, so the limit is the ratio of leading coefficients: 3 / 1.
3. Conclusion: the limit is 3 as x → ∞ (and, by the same reasoning, also 3 as x → -∞, since even-degree leading terms behave the same in both directions).

x → ∞ versus x → -∞: when the answer differs by direction

For an even-degree leading term (x², x⁴, and so on), the sign of x barely matters once it's squared, so the limit is usually the same heading toward +∞ and -∞. For an odd-degree leading term (x, x³, x⁵...), the sign absolutely matters: x³ heads toward +∞ as x → ∞, but toward -∞ as x → -∞. This calculator evaluates the two directions completely independently, which is exactly why it is worth double-checking both when a function has an odd-degree dominant term, rather than assuming the answer is the same in both directions.

Non-polynomial functions at infinity

Not every "limit at infinity" problem is a ratio of polynomials. For exponential, logarithmic, or trigonometric functions, the calculator falls back to a numerical check: sampling f(x) at very large values of x (and very negative values, for x → -∞) and checking whether the values settle toward a stable number, grow without bound, or oscillate. A classic example is limx→∞ e-x, which decays toward 0 as x grows (exponential decay always beats polynomial growth), while limx→∞ ln(x) grows without bound, just extremely slowly, so it is reported as +∞, not as a finite convergence.

Horizontal asymptotes and limits at infinity are the same question

If limx→∞ f(x) = L (a finite number), then the line y = L is a horizontal asymptote of the graph of f, the value the curve flattens out toward on the right side of the graph. A function can have a different horizontal asymptote on the left (as x → -∞) than on the right, or one side only, or none at all if the limit in that direction is infinite or does not settle. Calculating a limit at infinity is therefore exactly the calculation behind finding a function's horizontal asymptotes, just phrased as a graphing question instead of a symbolic one.

Related tools

For a limit at a finite point rather than at infinity, or for one-sided limits and factoring examples, see the general-purpose limit calculator. For an indeterminate ∞/∞ form specifically (for example, an exponential over a polynomial, where degree comparison does not directly apply), the L'Hopital's rule calculator differentiates numerator and denominator symbolically instead.