To determine which expression approximates the mathematical constant [tex]\( e \)[/tex] as [tex]\( n \)[/tex] becomes very large, let's evaluate each of the given options step-by-step.
### Option A: [tex]\(\left(1+\frac{1}{n}\right)^n\)[/tex]
This expression is known to approximate the value of [tex]\( e \)[/tex]. As [tex]\( n \)[/tex] approaches infinity:
[tex]\[
\left(1+\frac{1}{n}\right)^n \approx e
\][/tex]
Calculating this for a large value of [tex]\( n \)[/tex]:
[tex]\[
n = 1{,}000{,}000 \quad \Rightarrow \quad \left(1+\frac{1}{1{,}000{,}000}\right)^{1{,}000{,}000}
\][/tex]
The numerical approximation of this expression is approximately:
[tex]\[
\left(1+\frac{1}{1{,}000{,}000}\right)^{1{,}000{,}000} \approx 2.7182804690957534
\][/tex]
### Option B: [tex]\(\left(1-\frac{1}{n}\right)^n\)[/tex]
Let's analyze this expression. As [tex]\( n \)[/tex] gets very large, this expression approaches:
[tex]\[
\left(1-\frac{1}{n}\right)^n
\][/tex]
This does not converge to [tex]\( e \)[/tex]; rather, it converges to a limit less than 1:
[tex]\[
\left(1-\frac{1}{n}\right)^n \to \frac{1}{e} \quad \text{as} \quad n \to \infty
\][/tex]
### Option C: [tex]\((1+n)^{1 / n}\)[/tex]
This expression also diverges from [tex]\( e \)[/tex] as [tex]\( n \)[/tex] approaches infinity. Evaluating the behavior of the function:
[tex]\[
(1+n)^{1 / n} \quad \text{as} \quad n \to \infty
\][/tex]
Yields:
[tex]\[
(1+n)^{1 / n} \to 1 \quad \text{as} \quad n \to \infty
\][/tex]
### Option D: [tex]\((1-n)^{1 / n}\)[/tex]
For large [tex]\( n \)[/tex], this expression becomes mathematically unstable since it involves raising a negative number to a fraction, especially when [tex]\( n \)[/tex] is an integer:
[tex]\[
(1-n)^{1 / n}
\][/tex]
This is undefined for large [tex]\( n \)[/tex] as the base [tex]\( (1 - n) \)[/tex] is negative and raising it to the power of [tex]\( 1/n \)[/tex] doesn’t provide a meaningful real number result.
Given the above analysis, the correct expression to approximate [tex]\( e \)[/tex] is:
[tex]\[
\boxed{\left(1+\frac{1}{n}\right)^n}
\][/tex]
