Which of the following value is just greater than `[1+1/(10^100)]^(10^100)`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem of finding which of the following values is just greater than \(\left(1 + \frac{1}{10^{100}}\right)^{10^{100}}\), we can follow these steps:

### Step 1: Rewrite the expression
Let \( n = 10^{100} \). Then, we can rewrite the expression as:
\[ \left(1 + \frac{1}{n}\right)^n \]

### Step 2: Use the limit definition of \( e \)
As \( n \) approaches infinity, we know from calculus that:
\[ \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e \]
where \( e \) is Euler's number, approximately equal to \( 2.71828 \).

### Step 3: Analyze the expression for large \( n \)
Since \( n = 10^{100} \) is a very large number, we can conclude that:
\[ \left(1 + \frac{1}{10^{100}}\right)^{10^{100}} \approx e \]

### Step 4: Determine the value of \( e \)
The value of \( e \) is approximately \( 2.718 \). Thus, we need to find the smallest integer greater than \( e \).

### Step 5: Compare with given options
The options provided are:
1. 1
2. 2
3. 3
4. 4

Since \( e \approx 2.718 \), the smallest integer greater than \( e \) is \( 3 \).

### Conclusion
Thus, the value that is just greater than \(\left(1 + \frac{1}{10^{100}}\right)^{10^{100}}\) is:
\[ \boxed{3} \]