Infinity

Q: What is infinity minus infinity?

The expression $\infty - \infty$ is an indeterminate form — it does not equal zero or any other specific value. Different situations involving $\infty - \infty$ can produce different results. For instance, $\lim_{x \to \infty}(x - x) = 0$, but $\lim_{x \to \infty}(x^2 - x) = \infty$. You need more context (usually a limit) to determine the actual value.

Infinity

A "number" which indicates a quantity, size, or magnitude that is larger than any real number. The number infinity is written as a sideways eight: ∞. Negative infinity is written –∞.

Note: Neither ∞ nor –∞ is a real number.

See also

Infinite, finite, infinitesimal

Example

Problem: Evaluate the limit: What happens to 1/x as x gets closer and closer to 0 from the positive side?

Step 1: Substitute values of x that approach 0 from the right (positive side) and observe the output.

x = 1 \implies \frac{1}{x} = 1

Step 2: Try a smaller value.

x = 0.01 \implies \frac{1}{x} = 100

Step 3: Try an even smaller value.

x = 0.0001 \implies \frac{1}{x} = 10{,}000

Step 4: As x shrinks toward 0, the output grows without bound. No matter how large a number you pick, 1/x will eventually exceed it. We express this using the infinity symbol.

\lim_{x \to 0^+} \frac{1}{x} = +\infty

Answer: The function 1/x increases without bound as x approaches 0 from the positive side, so the limit equals positive infinity.

Another Example

Problem: Evaluate the limit of (3x + 1) / (x + 2) as x approaches infinity.

Step 1: Divide every term in the numerator and denominator by x, the highest power of x in the denominator.

\frac{3x + 1}{x + 2} = \frac{3 + \frac{1}{x}}{1 + \frac{2}{x}}

Step 2: As x approaches infinity, the fractions 1/x and 2/x both approach 0.

\lim_{x \to \infty} \frac{1}{x} = 0 \quad \text{and} \quad \lim_{x \to \infty} \frac{2}{x} = 0

Step 3: Substitute these limiting values into the simplified expression.

\frac{3 + 0}{1 + 0} = 3

Answer: The limit is 3. Even though x grows without bound, the ratio settles at the finite value 3.

Frequently Asked Questions

Is infinity a number?

Infinity is not a real number. You cannot place it on the number line alongside numbers like 2 or −7. It is a concept describing unbounded growth or size. Some extended number systems do treat infinity as a formal element, but in standard arithmetic and algebra, infinity does not obey the usual rules of numbers (for example,

\infty - \infty

is undefined).

What is infinity minus infinity?

The expression

\infty - \infty

is an indeterminate form — it does not equal zero or any other specific value. Different situations involving

\infty - \infty

can produce different results. For instance,

\lim_{x \to \infty}(x - x) = 0

, but

\lim_{x \to \infty}(x^2 - x) = \infty

. You need more context (usually a limit) to determine the actual value.

Infinity (∞) vs. Undefined

Infinity describes a quantity that grows without bound in a specific direction (positive or negative). 'Undefined' means an expression has no meaningful value at all — for example, 0/0 is undefined because no single number works. A limit can equal infinity (meaning the output grows endlessly), but an expression like 0/0 is undefined because it lacks a determinable result. Infinity tells you something about behavior; undefined tells you the expression breaks down.

Why It Matters

Infinity is the foundation of calculus. Limits, derivatives, and integrals all depend on the idea of quantities approaching infinity or becoming infinitely small. Beyond calculus, infinity appears in set theory (comparing sizes of infinite sets), physics (singularities, cosmology), and computer science (analyzing algorithm behavior as input size grows without bound).

Common Mistakes

Mistake: Treating infinity as a real number and performing ordinary arithmetic with it, such as writing

\infty + 1 = \infty

and then 'canceling' to get

1 = 0

Correction: Infinity does not follow the same algebraic rules as real numbers. You cannot add, subtract, multiply, or divide with

\infty

the way you do with finite numbers. Always work with limits instead of plugging

\infty

directly into expressions.

Mistake: Assuming

\infty - \infty = 0

\frac{\infty}{\infty} = 1

Correction: Both

\infty - \infty

and

\frac{\infty}{\infty}

are indeterminate forms. Their actual values depend on the specific functions involved. Use techniques like L'Hôpital's rule or algebraic simplification to evaluate the corresponding limit.

Related Terms

Real Numbers — The number system where infinity is excluded
Infinite — Adjective describing unbounded sets or processes
Finite — Opposite concept: bounded or limited quantity
Infinitesimal — Infinitely small counterpart to infinity
Limit — Primary tool for rigorously handling infinity
Asymptote — Line a graph approaches as x or y tends to infinity
Indeterminate — Forms like ∞−∞ that need further analysis
Diverge — Describes sequences or series that grow to infinity