Animated Solution for Mathematics - Limits, Continuity and Differentiability: The positive integer just greater than $(1+0.0001{)}^{10000}$ is

Imagine you are standing before a mountain of numbers. You see the expression $(1+0.0001{)}^{10000}$ and your first instinct might be to panic.

How on earth are we supposed to calculate that? Is it $2$? Is it $3$? Is it something else entirely?

My dear student, take a deep breath. In the world of JEE Advanced, we never solve by brute force. We solve by insight. This problem is not a calculation; it is a riddle waiting to be decoded.

Let us look at the decimal $0.0001$. In the language of fractions, this is simply $\frac{1}{10000}$.

Now, look at the exponent. It is also $10000$. Do you see the symmetry?

We have transformed our intimidating expression into:

This is the exact structural signature of the fundamental limit definition of Euler's number, $e$. We define $e$ as the limit of the sequence $a_n=(1+\frac{1}{n}{)}^n$ as $n$ approaches infinity.

By substituting $n=10000$, we are essentially looking at a specific snapshot of this sequence as it marches toward its destiny.

Now, here is where the magic happens. We know that as $n\to \infty$, the sequence $(1+\frac{1}{n}{)}^n$ converges to $e$, which is approximately $2.71828$.

But does it approach $e$ from above or below? This is the crucial realization.

Through the lens of calculus—specifically by examining the derivative of the function $f(x)=(1+\frac{1}{x}{)}^x$—we can prove that this sequence is strictly increasing. It is a monotonic climb.

It starts at $n=1$ (where the value is $2$) and it grows, step by step, getting closer and closer to that invisible ceiling of $e$. Because it is strictly increasing and bounded by $e$, every single finite term of this sequence must be strictly less than $e$.

So, where does that leave us? We have established that for $n=10000$, our value is strictly less than $e$.

Since $e\approx 2.718$, we know that:

Now, look at the number line. We have a value that is greater than $2$ but less than $2.718$.

The question asks for the positive integer just greater than this value. If you are standing at $2.718$ on the number line, the next integer to your right is $3$.

It is not $2$, because our value is already larger than $2$. It is not $4$, because that would be skipping over $3$.

The answer, elegant and simple, is $3$.

You see? We didn't need a calculator. We needed only the courage to look at the structure, the patience to identify the limit, and the logic to place our result on the number line. Mathematics is not about the destination; it is about the journey of understanding the patterns that govern our universe.