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Limits at Infinity — Definition, Formula & Examples

A limit at infinity describes the value that a function approaches as the input xx increases or decreases without bound. If f(x)f(x) gets closer and closer to a single number LL as x→∞x \to \infty or x→−∞x \to -\infty, then LL is the limit at infinity.

Let ff be a function defined on an interval (a,∞)(a, \infty). We write lim⁡x→∞f(x)=L\lim_{x \to \infty} f(x) = L if for every ε>0\varepsilon > 0 there exists a real number MM such that whenever x>Mx > M, it follows that ∣f(x)−L∣<ε|f(x) - L| < \varepsilon. An analogous definition holds for x→−∞x \to -\infty. When such a finite limit LL exists, the line y=Ly = L is a horizontal asymptote of ff.

Key Formula

lim⁡x→∞anxn+⋯+a0bmxm+⋯+b0={0if n<manbmif n=m±∞if n>m\lim_{x \to \infty} \frac{a_n x^n + \cdots + a_0}{b_m x^m + \cdots + b_0} = \begin{cases} 0 & \text{if } n < m \\ \dfrac{a_n}{b_m} & \text{if } n = m \\ \pm\infty & \text{if } n > m \end{cases}
Where:
  • ana_n = Leading coefficient of the numerator polynomial
  • bmb_m = Leading coefficient of the denominator polynomial
  • nn = Degree of the numerator polynomial
  • mm = Degree of the denominator polynomial

How It Works

To evaluate a limit at infinity for a rational function (a polynomial divided by a polynomial), compare the degrees of the numerator and denominator. If the degree on top is less than the degree on the bottom, the limit is 00. If the degrees are equal, the limit is the ratio of the leading coefficients. If the top degree exceeds the bottom degree, the function grows without bound and no finite limit exists. For non-rational functions, you can often divide by the dominant term or apply L'Hôpital's Rule when the expression takes an indeterminate form like ∞∞\frac{\infty}{\infty}.

Worked Example

Problem: Find lim⁡x→∞4x2+32x2−5\lim_{x \to \infty} \dfrac{4x^2 + 3}{2x^2 - 5}.
Step 1: Identify the degrees. Both the numerator and denominator are degree 2, so the degrees are equal.
n=2,m=2n = 2, \quad m = 2
Step 2: Since the degrees match, the limit equals the ratio of the leading coefficients.
anbm=42=2\frac{a_n}{b_m} = \frac{4}{2} = 2
Step 3: You can verify by dividing every term by x2x^2, the highest power in the denominator.
lim⁡x→∞4+3x22−5x2=4+02−0=2\lim_{x \to \infty} \frac{4 + \frac{3}{x^2}}{2 - \frac{5}{x^2}} = \frac{4 + 0}{2 - 0} = 2
Answer: lim⁡x→∞4x2+32x2−5=2\lim_{x \to \infty} \dfrac{4x^2 + 3}{2x^2 - 5} = 2, so y=2y = 2 is a horizontal asymptote.

Another Example

Problem: Find lim⁡x→−∞6xx2+1\lim_{x \to -\infty} \dfrac{6x}{x^2 + 1}.
Step 1: The numerator has degree 1 and the denominator has degree 2. Since the numerator degree is smaller, the limit is 0.
n=1<m=2n = 1 < m = 2
Step 2: Confirm by dividing every term by x2x^2.
lim⁡x→−∞6x1+1x2=01+0=0\lim_{x \to -\infty} \frac{\frac{6}{x}}{1 + \frac{1}{x^2}} = \frac{0}{1 + 0} = 0
Answer: lim⁡x→−∞6xx2+1=0\lim_{x \to -\infty} \dfrac{6x}{x^2 + 1} = 0, confirming y=0y = 0 as a horizontal asymptote.

Visualization

Why It Matters

Limits at infinity are tested heavily on the AP Calculus AB and BC exams, especially when identifying horizontal asymptotes and analyzing end behavior of functions. In engineering and physics, they model steady-state behavior — for instance, the terminal velocity an object approaches as time grows large. Mastering this concept also prepares you for improper integrals and series convergence in Calculus II.

Common Mistakes

Mistake: Confusing a limit at infinity with an infinite limit. Students write lim⁡x→∞f(x)=∞\lim_{x \to \infty} f(x) = \infty and call it a 'limit at infinity equaling infinity.'
Correction: A limit at infinity asks what happens as xx grows without bound. An infinite limit means the function's output blows up (often near a vertical asymptote). When the result is ±∞\pm\infty, the finite limit at infinity does not exist — though we still describe the end behavior.
Mistake: Dividing by the wrong power of xx when simplifying a rational function.
Correction: Always divide every term in both the numerator and denominator by the highest power of xx that appears in the denominator. This ensures the denominator approaches a finite nonzero constant.

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