Infinity Formula

The Formula

\lim_{x \to \infty} \frac{1}{x^p} = 0 \text{ for } p > 0 \qquad \lim_{x \to 0^+} \frac{1}{x^p} = +\infty \text{ for } p > 0

When to use: Going on forever without end. Infinity is a direction or limiting idea, not a number you can reach or write down.

Quick Example

\lim_{x \to \infty} \frac{1}{x} = 0 As x gets arbitrarily large, \frac{1}{x} approaches 0.

Notation

\infty (infinity), -\infty (negative infinity). x \to \infty means x grows without bound.

What This Formula Means

A concept representing a quantity that grows without bound — infinity is not a real number but a description of unbounded behavior.

Going on forever without end. Infinity is a direction or limiting idea, not a number you can reach or write down.

Formal View

\lim_{x \to \infty} f(x) = L \iff \forall \epsilon > 0,\; \exists M > 0 : x > M \implies |f(x) - L| < \epsilon. The limit equals \infty: \lim_{x \to a} f(x) = \infty \iff \forall N > 0,\; \exists \delta > 0 : 0 < |x - a| < \delta \implies f(x) > N.

Worked Examples

Example 1

easy
Evaluate \lim_{x \to \infty} \frac{3x^2 + 5}{x^2 - 1}.

Solution

  1. 1
    Divide numerator and denominator by the highest power of x in the denominator, x^2.
  2. 2
    Numerator: \frac{3x^2 + 5}{x^2} = 3 + \frac{5}{x^2}. Denominator: \frac{x^2-1}{x^2} = 1 - \frac{1}{x^2}.
  3. 3
    As x \to \infty, \frac{5}{x^2} \to 0 and \frac{1}{x^2} \to 0.
  4. 4
    Limit: \frac{3 + 0}{1 - 0} = 3.

Answer

3
For rational functions at infinity, divide by the highest power of x in the denominator. Terms with x in the denominator vanish, leaving only the ratio of leading coefficients. When degrees are equal, the limit is the ratio of leading coefficients.

Example 2

medium
Evaluate \lim_{x \to \infty} \frac{2x^3 - x}{5x^2 + 3}.

Common Mistakes

  • Treating \infty as a real number and performing arithmetic: \infty - \infty \neq 0 and \frac{\infty}{\infty} \neq 1 — these are indeterminate forms that require careful limit analysis.
  • Assuming \frac{1}{0} = \infty without considering the sign: \lim_{x \to 0^+} \frac{1}{x} = +\infty but \lim_{x \to 0^-} \frac{1}{x} = -\infty, so the two-sided limit does not exist.
  • Confusing 'limit equals infinity' with 'limit exists': saying \lim_{x \to 0} \frac{1}{x^2} = \infty means the limit does NOT exist as a finite number — it describes unbounded behavior.

Why This Formula Matters

Understanding infinity is essential for limits, asymptotes, convergence of series, and improper integrals.

Frequently Asked Questions

What is the Infinity formula?

A concept representing a quantity that grows without bound — infinity is not a real number but a description of unbounded behavior.

How do you use the Infinity formula?

Going on forever without end. Infinity is a direction or limiting idea, not a number you can reach or write down.

What do the symbols mean in the Infinity formula?

\infty (infinity), -\infty (negative infinity). x \to \infty means x grows without bound.

Why is the Infinity formula important in Math?

Understanding infinity is essential for limits, asymptotes, convergence of series, and improper integrals.

What do students get wrong about Infinity?

\infty - \infty is undefined, not 0. \frac{\infty}{\infty} is undefined. These are 'indeterminate forms.'

What should I learn before the Infinity formula?

Before studying the Infinity formula, you should understand: limit.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Limits Explained Intuitively: The Foundation of Calculus →
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