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You can approximate [tex]$e$[/tex] by substituting large values for [tex]$n$[/tex] into the expression:

A. [tex]\left(1+\frac{1}{n}\right)^n[/tex]

B. [tex]\left(1-\frac{1}{n}\right)^n[/tex]

C. [tex](1+n)^{1/n}[/tex]

D. [tex](1-n)^{1/n}[/tex]


Sagot :

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]