Limits at Infinity: Step-by-Step Guide
A limit at infinity describes the end behavior of a function as x grows without bound (positive or negative). For rational functions, the answer depends on comparing the degrees of the numerator and denominator; for radicals and other forms, divide by the highest power to reduce to known limits like 1/x -> 0.
Math Racer skill guide · Calculus 1 · AP Calc AB / College
How to solve it, step by step
- Identify the highest power of x. Find the largest exponent of x appearing anywhere in the expression (in either numerator or denominator).\[n = \max(\text{degree of numerator}, \text{degree of denominator})\]
- Divide every term by x^n. Divide numerator and denominator by x^n. Each term becomes a constant or a fraction whose denominator grows without bound.\[\frac{f(x)}{g(x)} = \frac{f(x)/x^n}{g(x)/x^n}\]
- Apply lim x->infty (1/x^k) = 0 for k > 0. Every term of the form c/x^k (with k > 0) goes to 0 as x -> infinity. Replace such terms with 0 and simplify.\[\lim_{x \to \infty} \frac{1}{x^k} = 0, \;\; k > 0\]
- Read off the result. If degrees are equal, the limit is the ratio of leading coefficients. If numerator degree is smaller, the limit is 0. If numerator degree is larger, the limit is plus or minus infinity (sign depends on leading coefficients).\[\lim_{x \to \infty} \frac{a_n x^n + \dots}{b_m x^m + \dots}\]
Worked example
Evaluate lim x->infty (3x^2 + 2x - 1)/(5x^2 - x + 4).
Step 1: Highest power of x is 2 (both numerator and denominator are degree 2).
Step 2: Divide every term by x^2: ((3x^2 + 2x - 1)/x^2)/((5x^2 - x + 4)/x^2) = (3 + 2/x - 1/x^2)/(5 - 1/x + 4/x^2).
Step 3: Apply 1/x -> 0 and 1/x^2 -> 0: numerator becomes 3 + 0 - 0 = 3 and denominator becomes 5 - 0 + 0 = 5.
Step 4: The limit equals 3/5.
Step 5: This matches the shortcut: equal degrees imply the limit is the ratio of leading coefficients, 3/5.
Verification: At x = 1000, numerator approximately 3,002,000 and denominator approximately 4,999,004; ratio approximately 0.6005, very close to 0.6 = 3/5.
Tips and common mistakes
- For rational functions, compare degrees first: equal -> ratio of leading coefficients; top smaller -> 0; top larger -> infinite.
- Lower-order terms become negligible at infinity; only the leading terms matter for the final value.
- When x -> -infinity, watch the sign carefully, especially with odd powers and square roots.
- For sqrt(x^2 + ...) treat sqrt(x^2) as |x|, which equals x when x -> +infinity but -x when x -> -infinity.
- Exponentials grow faster than any polynomial, and polynomials grow faster than any logarithm.
- If you see infinity - infinity, infinity*0, or similar indeterminate forms, manipulate (factor, common denominator, conjugate) before taking the limit.
- Horizontal asymptotes correspond exactly to finite limits at plus or minus infinity.
- Always sanity-check by plugging in a very large x (like 10^6) and comparing numerically.