What Happens When Infinity Is Multiplied by an Infinitesimal?

The phrase “infinity times an infinitesimal” can refer to a limit in ordinary calculus or to a product of special numbers in nonstandard analysis. The answer depends on which meaning you intend. In the common calculus setting, the product has no predetermined value: you must compare how fast one factor grows and the other shrinks.

In calculus, it is an indeterminate form

In ordinary real-number arithmetic, infinity is not a real number, and an infinitesimal is not a nonzero real number. Instead, the symbols usually summarize behavior: one quantity grows without bound while another tends to zero. The notation $\infty\cdot 0$ is therefore shorthand for a limit with those behaviors, not an arithmetic operation with a fixed answer.

That limit is called an indeterminate form. “Indeterminate” does not mean that the limit cannot be found. It means that the information “one factor goes to infinity and the other goes to zero” is not enough to determine the result. The relative rates matter.

Compare the rates with examples

Let $x$ approach zero through positive values. In each example, the first factor tends to infinity and the second tends to zero:

$$\begin{aligned} \frac{1}{x}\,x &= 1,\\ \frac{1}{x}\,x^2 &= x\longrightarrow 0,\\ \frac{1}{x}\,\sqrt{x} &= \frac{1}{\sqrt{x}}\longrightarrow \infty. \end{aligned}$$

So the same general pattern—an unbounded factor multiplied by a vanishing factor—can produce a limit of $1$, $0$, or infinity. Other outcomes are possible too. For example, $\frac{2}{x}\cdot x$ tends to $2$, and a negative factor can give a negative finite limit or negative infinity.

The useful question is not simply “What is infinity times zero?” but “How does the product behave after I express both factors in terms of the same variable?”

A practical method for evaluating the limit

Suppose you need to find

$$\lim_{x\to a} f(x)g(x),$$

where $f(x)$ grows without bound and $g(x)$ tends to zero. Try these steps:

1. Write the product as one expression in the variable, then simplify or factor it.
2. Compare the dominant terms near $a$. For powers, the exponents often reveal which factor wins.
3. Rewrite as a quotient if useful. For instance, $f(x)g(x)=f(x)/(1/g(x))$ when $g(x)\ne 0$. This may let you apply a familiar limit rule or, when its conditions hold, L’Hôpital’s rule.
4. Check signs and the direction of approach. A one-sided limit may differ from the other side, and the product could tend to positive or negative infinity.

For a concrete comparison, consider $x\to\infty$ and the product $x^p x^{-q}=x^{p-q}$, where $p,q>0$. If $p<q$, it tends to zero; if $p=q$, it equals $1$; if $p>q$, it tends to infinity. The exponents capture the balance between growth and decay.

What if “infinity” and “infinitesimal” are actual numbers?

Nonstandard analysis extends the real numbers to include infinite and infinitesimal numbers. There, ordinary algebraic multiplication applies, but the product still depends on the sizes of the factors.

For example, let $H$ be a positive infinite hyperinteger. Then $1/H$ is a positive infinitesimal, and

$$H\cdot\frac{1}{H}=1,\qquad H\cdot\frac{1}{H^2}=\frac{1}{H},\qquad H\cdot\frac{1}{\sqrt{H}}=\sqrt{H}.$$

The three products are finite, infinitesimal, and infinite, respectively. Thus even when these terms name numbers rather than limit behavior, “infinite times infinitesimal” does not specify one result; the relative magnitudes are still needed.

Common mistakes

• Treating $\infty$ as an ordinary real number. It is usually a limit description, not a value you can substitute into normal arithmetic.
• Assuming the product must be $1$. That is true for particular reciprocal pairs, such as $x(1/x)$, but not for every factor tending to zero and every factor growing without bound.
• Using L’Hôpital’s rule on the product directly. Its standard forms are quotient forms such as $0/0$ or $\infty/\infty$. If you rewrite the product as a quotient, verify that the hypotheses for the rule apply.
• Ignoring signs or one-sided behavior. For example, $1/x$ tends to opposite infinities on opposite sides of zero, so the direction of approach can change the conclusion.