52989712
A student is studying \(\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n\) and claims: “The expression inside the parentheses approaches \(1\), and \(1\) raised to an infinite power is \(1\), so the limit must be \(1\).”
1. Test the claim by calculating the expression for \(n=5\), \(n=50\), and \(n=500\). Round to three decimal places.
2. Explain why the student's reasoning is incomplete. Address how the base and exponent change together.
3. Name this limit and give its value rounded to two decimal places.
Hints
- Evaluate the expression for increasingly large values of \(n\).
- What happens when a number slightly greater than \(1\) is multiplied by itself many times?
- Are the computed values moving toward \(1\)?
- Consider whether separate limiting behaviors can compete with each other.
Solution
1. For \(n=5\), \(\left(1+\frac{1}{5}\right)^5\approx2.488\). For \(n=50\), \(\left(1+\frac{1}{50}\right)^{50}\approx2.692\). For \(n=500\), \(\left(1+\frac{1}{500}\right)^{500}\approx2.716\).
2. The base approaches \(1\), but the exponent grows without bound at the same time. A number slightly greater than \(1\), multiplied by itself an increasing number of times, need not approach \(1\). The form \(1^{\infty}\) is indeterminate, so the combined limit must be analyzed rather than found by substituting the two separate limits.
3. The limit is Euler's number \(e\), and \(e\approx2.72\).
Answer
1. \(2.488\), \(2.692\), and \(2.716\)
2. The base and exponent change simultaneously; \(1^{\infty}\) is an indeterminate form.
3. Euler's number, \(e\approx2.72\)
