What does the e in e \approx 2.718 have to do with compound interest?

The limit definition shows that e naturally arises when you compound interest infinitely often. If you invest 1 dollar at 100% annual interest compounded n times per year, as n \to \infty your return approaches e dollars. This is why e is the base of natural exponential growth and decay in finance, biology, and physics.

Why do I need x \to 0 in the exponent and not n \to \infty?

Both are valid—the definitions \lim_{x \to 0} (1 + x)^{1/x} = e and \lim_{n \to \infty} (1 + 1/n)^n = e are equivalent. Use whichever substitution makes your problem simpler. If your problem naturally has a term going to 0, use the first; if it naturally has a term going to infinity, use the second.

Can I use L'Hopital's Rule instead of the limit definition of e?

Yes. Since (1 + x)^{1/x} has the indeterminate form 1^{\infty} as x \to 0, you can rewrite it as e^{\ln(1 + x)/x} and apply L'Hopital's Rule to the exponent. However, the limit definition of e is more direct and shows why e is defined this way in the first place.

What if the coefficient in the base is not 1, like (1 + 3x)^{1/x}?

Use substitution to rewrite it in the standard form. If u = 3x, then (1 + 3x)^{1/x} = (1 + u)^{3/u} = [(1 + u)^{1/u}]^3 \to e^3. The key is ensuring both the base coefficient and the exponent coefficient match so you can apply the definition.

Limit Definition of e

The limit definition of e shows how the mathematical constant e arises from the growth of compound interest and exponential functions.

\lim_{x \to 0} (1 + x)^{1/x} = e

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What each symbol means

What Limit Definition of e takes

Symbol	Meaning
$x$	x is a dimensionless variable approaching 0 in both the base $1+x$ and the exponent $1/x$ ; misreading one of these roles breaks the entire formula.
$e$	e is the mathematical constant approximately 2.71828, the base of natural logarithms; confusing it with a different constant like 10 or with a variable $e$ used elsewhere corrupts the result.

When to use it

Use this definition when evaluating limits of the form $(1 + x)^{1/x}$ or when proving properties of the exponential function.

Level

Usually taught in: Calculus I

Worked examples

1. Evaluate a limit with a scaled base

Problem

Evaluate

\lim_{x \to 0} (1 + 2x)^{1/x}

$Let \quad u = 2x$
We set up a substitution so that as $x \to 0$ , we have $u \to 0$ , and the base becomes $1 + u$ .
$x = \frac{u}{2} \quad \text{and} \quad \frac{1}{x} = \frac{2}{u}$
We solve for $x$ and $1/x$ in terms of $u$ to rewrite the exponent.
$(1 + 2x)^{1/x} = (1 + u)^{2/u} = [(1 + u)^{1/u}]^2$
After substitution, we factor the exponent so that $(1 + u)^{1/u}$ is isolated, matching the limit definition form.
$\lim_{x \to 0} (1 + 2x)^{1/x} = \lim_{u \to 0} [(1 + u)^{1/u}]^2 = e^2$
Taking the limit as $u \to 0$ , the expression $(1 + u)^{1/u}$ approaches $e$ , so the entire limit is $e^2$ .

Answer: $e^2$

The key insight is recognizing that $(1 + 2x)^{1/x}$ can be rewritten using substitution to match the standard form $(1 + u)^{1/u}$ . When the exponent is a constant multiple of $1/u$ like $2/u$ , we can factor it out and apply the limit definition to the core $(1 + u)^{1/u}$ term, then raise the result to the appropriate power.

2. Evaluate a limit with a negative coefficient

Problem

Evaluate

\lim_{x \to 0} (1 - 3x)^{1/x}

$Let \quad u = -3x$
We substitute so that the base $(1 - 3x)$ becomes $(1 + u)$ , which matches the standard form.
$x = -\frac{u}{3} \quad \text{and} \quad \frac{1}{x} = -\frac{3}{u}$
Solving for $x$ and $1/x$ in terms of $u$ gives us the exponent in the new variable.
$(1 - 3x)^{1/x} = (1 + u)^{-3/u} = [(1 + u)^{1/u}]^{-3}$
After substitution, the exponent is $-3/u$ , and factoring reveals that the negative sign is preserved.
$\lim_{x \to 0} (1 - 3x)^{1/x} = \lim_{u \to 0} [(1 + u)^{1/u}]^{-3} = e^{-3}$
The core limit is $e$ , and we raise it to the power of $-3$ to get $e^{-3}$ , which is the reciprocal of $e^3$ .

Answer: $e^{-3}$

When the base has a negative coefficient, the substitution must convert the minus into a plus, which introduces a negative factor in the exponent. The negative sign carries through to the final answer as a negative exponent, illustrating how the limit definition handles both growth $(e^k)$ and decay $(e^{-k})$ patterns.

3. Model cooling behavior in a thermal lab

Problem

A thermal testing lab models the cooling of a heated object using the formula

(1 - 0.2t)^{1/(0.2t)}

where

t

is time in seconds. As the measurement interval shrinks to 0, find the limiting cooling factor.

$Let \quad u = -0.2t$
We substitute to convert $(1 - 0.2t)$ into the standard form $(1 + u)$ .
$t = -5u \quad \text{and} \quad \frac{1}{0.2t} = -\frac{1}{u}$
Solving for $t$ and the reciprocal of $0.2t$ in terms of $u$ allows us to rewrite the exponent.
$(1 - 0.2t)^{1/(0.2t)} = (1 + u)^{-1/u} = [(1 + u)^{1/u}]^{-1}$
After substitution and factoring, we isolate the $(1 + u)^{1/u}$ term with a negative exponent.
$\lim_{t \to 0} (1 - 0.2t)^{1/(0.2t)} = \lim_{u \to 0} [(1 + u)^{1/u}]^{-1} = e^{-1} = \frac{1}{e}$
As $u \to 0$ , the limit is $e^{-1}$ , which is the reciprocal of $e$ , showing that the cooling factor approaches $1/e$ .

Answer: $e^{-1} \text{ or } \frac{1}{e}$

This problem shows how the limit definition of $e$ naturally appears in real-world cooling models. The fact that the limiting cooling factor is exactly $1/e$ demonstrates that the mathematical constant $e$ is not arbitrary but emerges from fundamental exponential processes in nature. By recognizing the form and applying substitution strategically, abstract calculus connects directly to measurable physical phenomena.

Common mistakes

Where Limit Definition of e usually goes wrong

The mistake	Why it is wrong	The fix
$\lim_{x \to 0} (1 + x)^{1/x} = 1$ (treating it like an indeterminate form that equals 1)	This is a $1^{\infty}$ indeterminate form, but it does not collapse to 1; instead it has the specific non-trivial limit value of $e \approx 2.718$ .	Recognize that $1^{\infty}$ is indeterminate and does not default to 1; the limit definition of $e$ proves that this limit equals exactly $e$ .
When evaluating $\lim_{x \to 0} (1 + 2x)^{1/x}$ , directly writing $(1 + 2x)^{1/x} = e^{2}$ without substitution.	The base is $(1 + 2x)$ , not $(1 + x)$ , so you cannot immediately apply the definition; you must rewrite it in the standard form first.	Use substitution $u = 2x$ to rewrite $(1 + 2x)^{1/x} = [(1 + u)^{1/u}]^2$ , then apply the definition to get $e^2$ .
Confusing $\lim_{x \to 0} (1 + x)^{1/x} = e$ with $\lim_{n \to \infty} (1 + 1/n)^n = e$ and mixing up which form applies to your problem.	These are two different (but equivalent) definitions of $e$ ; using the wrong substitution strategy or failing to recognize which form your problem takes leads to incorrect limits.	Match your substitution to the form in your problem: the first definition applies when $x \to 0$ with exponent $1/x$ ; the second applies when $n \to \infty$ with exponent $n$ .

Tips and when to use something else

Look for the pattern $(1 + \text{something})^{1/(\text{same something})}$ and use substitution to match the standard form before applying the limit definition.
If your limit has the form $(1 + x)^a$ where $a$ is a constant, do not use the limit definition of $e$ ; instead just evaluate $(1 + x)^a$ at $x = 0$ to get $1^a = 1$ .
Remember that the two definitions of $e$ are equivalent: $\lim_{x \to 0} (1 + x)^{1/x} = \lim_{n \to \infty} (1 + 1/n)^n = e$ ; choose whichever form makes your problem simpler.
When the exponent is a multiple of $1/x$ like $(1 + x)^{5/x}$ , factor it as $[(1 + x)^{1/x}]^5$ before taking the limit to isolate the definition.

Related formulas

Definition of a LimitDefinition of a Limit describes what value a function approaches as its input approaches a specific point—essential for all of calculus.Indeterminate FormsIndeterminate forms are expressions like 0/0 and ∞/∞ that arise when evaluating limits, signaling that direct substitution doesn't work.L'Hopital's RuleL'Hopital's Rule efficiently solves limits that reduce to 0/0 or ∞/∞ forms by taking derivatives of the numerator and denominator separately.Limits at InfinityFind what a rational function approaches as x grows indefinitely by comparing the degrees and leading coefficients of the numerator and denominator.Limit LawsThe Sum Law shows that a limit of a sum equals the sum of the limits, provided both limits exist and are finite numbers.

Frequently asked questions

What does the $e$ in $e \approx 2.718$ have to do with compound interest?: The limit definition shows that $e$ naturally arises when you compound interest infinitely often. If you invest 1 dollar at 100% annual interest compounded $n$ times per year, as $n \to \infty$ your return approaches $e$ dollars. This is why $e$ is the base of natural exponential growth and decay in finance, biology, and physics.
Why do I need $x \to 0$ in the exponent and not $n \to \infty$ ?: Both are valid—the definitions $\lim_{x \to 0} (1 + x)^{1/x} = e$ and $\lim_{n \to \infty} (1 + 1/n)^n = e$ are equivalent. Use whichever substitution makes your problem simpler. If your problem naturally has a term going to 0, use the first; if it naturally has a term going to infinity, use the second.
Can I use L'Hopital's Rule instead of the limit definition of $e$ ?: Yes. Since $(1 + x)^{1/x}$ has the indeterminate form $1^{\infty}$ as $x \to 0$ , you can rewrite it as $e^{\ln(1 + x)/x}$ and apply L'Hopital's Rule to the exponent. However, the limit definition of $e$ is more direct and shows why $e$ is defined this way in the first place.
What if the coefficient in the base is not 1, like $(1 + 3x)^{1/x}$ ?: Use substitution to rewrite it in the standard form. If $u = 3x$ , then $(1 + 3x)^{1/x} = (1 + u)^{3/u} = [(1 + u)^{1/u}]^3 \to e^3$ . The key is ensuring both the base coefficient and the exponent coefficient match so you can apply the definition.

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Reviewed 2026-09-18