Question

Evaluate the limit \[ \lim_{n \to \infty} \left[ n - \frac{n}{e} \left( 1 + \frac{1}{n} \right)^n \right] \text{ equals ............} \]

Hint:

When faced with the limit involving \( \left( 1 + \frac{1}{n} \right)^n \), recall that this expression converges to \( e \) as \( n \to \infty \).

Answer 1

Correct Answer: 0.5

Step 1: Simplify the expression.
We start by simplifying the expression inside the limit: \[ \left( 1 + \frac{1}{n} \right)^n \approx e \quad \text{as} \quad n \to \infty. \] This is a standard result from calculus, where the expression \( \left( 1 + \frac{1}{n} \right)^n \) approaches \( e \) as \( n \) becomes large.
Step 2: Substitute and simplify further.
Substituting \( \left( 1 + \frac{1}{n} \right)^n \approx e \) into the original expression: \[ n - \frac{n}{e} \times e = n - n = 0. \] Final Answer: \[ \boxed{0}. \]