When we first learn exponents, it seems straightforward:
should be infinitely large.
should be zero.- And since
, for all finite n, maybe 
Surprisingly, 
is not well-defined. In calculus, it’s called an indeterminate form. That means its value depends on the situation.
Understanding why requires tools from limits and sometimes L’Hôpital’s Rule. Let’s break it down.
Why is One to the Power of Infinity Indeterminate?
Infinity is not a number—it’s a concept. When you see something like , it usually comes from a limit where the base is approaching 1 while the exponent is growing without bound.
Example:
Here, the base (1 + x) approaches 1, while the exponent 1/x grows infinitely. That’s what creates the indeterminate form .
Indeterminate Forms Recap
In calculus, some expressions don’t immediately give clear answers. They’re called indeterminate forms, such as:
| Operation | Rule / Result | Notes |
|---|---|---|
| ∞ - ∞ | Indeterminate | Cannot be determined. |
| ∞ × 0 | Indeterminate | Undefined result. |
| ∞ ÷ ∞ | Indeterminate | Depends on context (limits). |
| 0^0 | Indeterminate | Sometimes treated as 1, sometimes undefined. |
| ∞^0 | Indeterminate | No fixed result. |
| 1^∞ | Indeterminate | Depends on the limit process. |
These require careful analysis with limits to find their true value.
L’Hôpital’s Rule is one of the most powerful tools for resolving indeterminate forms.
It states:
If
gives 
or 
, then:
provided the limit on the right exists.
How to Handle One to the Power of Infinity
Suppose:
gives .
Steps:
Take the natural logarithm:
Simplify the limit:
If it becomes 
or 
, apply L’Hôpital’s Rule.
Solve for ln(y)
Exponentiate to return to y
Example
Evaluate:
Step 1: Let
Take natural logs:
Step 2: Both numerator and denominator →0. Apply L’Hôpital’s Rule:
Step 3: Exponentiate to solve for y:
Practice Questions & Solutions
Evaluate:
Let
Take natural logs:
Apply L’Hôpital’s Rule:
Exponentiate:
Evaluate:
Let
Take natural logs:
As :
Denominator:
So the form is , perfect for L’Hôpital’s Rule.
Differentiate numerator and denominator (using the chain rule):
Therefore:
At x=0:
So:
Evaluate:
Let
Take natural logs:
For small x, use the expansion:
Thus,
So,
Exponentiate:
Summarise with AI:
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