What Is the Value of e Power Minus Infinity?

Chances are you ran into e^(−∞) in a calculus problem and weren’t sure what to do with it. Here’s the quick answer: the value of e power minus infinity is 0. The longer answer is more interesting. Infinity isn’t a number you can plug in, so e^(−∞) is really shorthand for a question: what happens to be raised to a power when that power keeps dropping further below zero? The result shrinks toward 0 and gets as close as you like, but it never dips below it. Let’s walk through why, one step at a time.

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A Quick Refresher on the Number e

The letter e stands for Euler’s number, roughly 2.71828. It’s irrational, so its decimals run on forever without a pattern, a bit like pi. You’ll see it in compound interest, population growth, radioactive decay and probability, which is why it gets so much attention in exams.

Only two facts about the function f(x) = eˣ matter for this topic:

It is always positive. Whatever real number you put in, eˣ comes out above zero.
It keeps climbing. Bigger x means bigger eˣ.

Hold on to those two points. The whole answer grows out of them.

Infinity Is a Direction, Not a Number

This is where many students get tripped up. You can’t treat ∞ like 5 or 100. It’s a way of saying “keeps growing with no upper limit.” So when a textbook writes e^(−∞), it’s compressing a longer sentence: “let the exponent become more and more negative and see where the value is heading.”

That’s exactly why serious maths writes this idea as a limit. Think of it as describing a journey instead of a destination you can stand on.

What Is e to the Power of Negative Infinity?

It equals 0. To see why, start small and watch the pattern:

Exponent	Value of e raised to it (approx.)
−1	0.3679
−5	0.006738
−10	0.0000454
−20	0.00000000206
−50	1.93 × 10⁻²²
−100	3.72 × 10⁻⁴⁴

Each time the exponent drops, the answer collapses. By −100 the value is so tiny that most calculators just show it in scientific notation. Keep going and the numbers keep sliding toward zero, which is what we mean when we say e to the power of negative infinity is 0.

What Is the Value of e^{-∞}?

There’s a second way to get the same answer, and many students find it easier to believe. A negative exponent means “flip it”:

e^{-\infty} = \frac{1}{e^{\infty}}

Now ask what e^∞ is. It’s enormous, with no ceiling at all. So you’re dividing 1 by something that grows without limit. Cut a cake into more and more slices and each slice gets smaller and smaller; that’s the same idea. A share that keeps shrinking heads for zero, and the value of e^{-∞} is 0.

Two roads, one answer. That’s usually a good sign in maths.

Evaluate the Limit of e^{-x} as x Approaches Infinity

Exam questions rarely say “e to the minus infinity.” They ask you to evaluate the limit of e^{-x} as x approaches infinity, and here’s how to write that up cleanly.

Rewrite the expression using the negative exponent rule: e⁻ˣ = 1/eˣ.
Look at the bottom of the fraction. As x → ∞, eˣ → ∞.
The top stays fixed at 1, so you have 1 divided by a quantity that grows without bound.
That gives 0.

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

The formal proof, if your syllabus asks for it

Higher-level courses want the epsilon argument. It sounds scary, but it’s short. Pick any tiny positive number ε less than 1. We want e⁻ˣ < ε.

Take the natural log of both sides. Since ln is increasing, and −x flips the inequality when we multiply by −1, we get x > ln(1/ε). So once x is bigger than N = ln(1/ε), the value of e⁻ˣ is below ε, and that holds no matter how small ε is. That is exactly the definition of the limit being 0.

What Does e Raised to Minus Infinity Equal?

In one word, zero. But it’s a particular kind of zero, and the difference shows up in exams.

For every real number x, e⁻ˣ is strictly bigger than 0. The function never lands on zero; it just keeps closing the gap. Picture walking toward a wall by covering half the remaining distance each step. You get closer forever, yet you never touch the wall. In graph language, the line y = 0 is a horizontal asymptote.

The graph of y = eˣ makes this easy to picture. On the right it shoots upward. On the left it flattens out and hugs the x-axis, and it passes through (0, 1) because e⁰ = 1. The graph of y = e⁻ˣ is its mirror image, sliding down toward the x-axis as you move right.

What is lim x→∞ e^{-x}?

That expression, lim x→∞ e^{-x}, is just the standard shorthand for the limit we solved above, read aloud as “the limit of e to the minus x, as x tends to infinity.” The answer is 0.

It’s worth knowing how this connects to two neighbours, because they get mixed up all the time:

lim (x→−∞) eˣ = 0. Same behaviour, written with the sign moved into x itself.
lim (x→∞) eˣ = ∞. Flip the sign of the exponent and the answer flips from tiny to huge.

So the sign of the exponent decides everything. Large and negative sends the value to 0; large and positive sends it to infinity.

Worked Examples

Example 1. Find lim (x→∞) 5e⁻²ˣ. As x grows, −2x heads to −∞, so e⁻²ˣ → 0. Multiply by 5 and you still get 0.

Example 2. Find lim (x→∞) (3 + e⁻ˣ). The e⁻ˣ part vanishes, leaving 3.

Example 3. Find lim (x→∞) x·e⁻ˣ. This looks like ∞ × 0, which isn’t an answer by itself. Rewrite it as x/eˣ, which is ∞/∞, and use L’Hôpital’s rule: the derivatives give 1/eˣ, which tends to 0. In plain words, exponential decay wins the tug of war against x.

Example 4. Find lim (x→∞) 1/(1 + e⁻ˣ). With e⁻ˣ → 0 you get 1/(1 + 0) = 1. This is the well-known sigmoid curve, which levels off at 1.

Try these yourself: lim (x→∞) 7e⁻ˣ, lim (x→∞) (4 − e⁻ˣ), and lim (x→∞) e^(−x²). The answers are 0, 4 and 0.

Mistakes That Cost Marks

Treating ∞ as an ordinary number. Don’t substitute it like you would 5. Think in limits.
Expecting a negative answer. The exponent is negative, but eˣ is always positive, so the result approaches 0 from above.
Mixing up e⁻ˣ and −eˣ. The first tends to 0 as x → ∞. The second heads off to −∞.
Saying it eventually equals 0. It gets arbitrarily close, but for any real x it is still a small positive number.
Skipping the extra step on products. Expressions like x·e⁻ˣ need L’Hôpital’s rule or a growth-rate argument before you can state the limit.

Where This Shows Up in Real Life

This limit isn’t just an exam trick. Anything that fades away exponentially relies on it. A radioactive sample follows N(t) = N₀e^(−λt), so the amount left tends to zero as time passes. A cup of tea cools toward room temperature the same way, a drug’s concentration in the blood drops after a dose, and a capacitor’s charge dies away as it discharges. In statistics, the tails of the normal and exponential distributions fall off using e⁻ˣ-style terms. In each case the fact that e⁻ˣ goes to 0 is what lets us say the effect eventually becomes negligible.

Conclusion

The value of e power minus infinity is 0. As the exponent sinks further below zero, e raised to it gets as close to zero as you like without ever turning negative. You can show it by rewriting e⁻ˣ as 1/eˣ, by watching the table of values, or by using the epsilon proof.

Frequently Asked Questions

Is e to the power of minus infinity exactly zero?

As a limit, yes. For any actual real input, though, the value is a tiny positive number and not exactly zero.

What is the power of infinity?

Infinity, because eˣ grows without bound as x increases.

What is the power of zero?

It’s 1, since any non-zero number raised to the power 0 equals 1.

Can a calculator confirm this?

It can support the idea. Try e^(−50) and you’ll see a very small number. Just remember that a calculator shows an example, while the limit is a proof about every large x.