Explain why $\lim ({a_n}^k)={(\lim(a_n))}^k$ cannot be used to find limit $(1+\frac1n)^n$

Question

Explain why lim $({a_n}^k)=(\lim(a_n))^k$ cannot be used to find limit $(1+\frac1n)^n$

The condition for this result to hold is $(a_n)$ has to be a convergent sequence and $k \in \mathbb N$, which are both true in this case, because lim $(1+\frac1n) =1$. So why can't we use this?

Thus $\lim\limits_{n\to\infty}\left(1+\frac1n\right)^k=1$ for every fixed $k$. How is this related to $\lim\limits_{n\to\infty}\left(1+\frac1n\right)^n$? — Did, May 03 '15 at 12:01
Here I try to explain the errors students are in the habit of making at this and similar other points. — Jyrki Lahtonen, May 03 '15 at 12:15

score 7 · Accepted Answer · answered May 03 '15 at 12:01

7

Because n is variable, while k is a constant in your property.

What means $(\lim_{n\to +\infty} a_n )^n$? $n$ is both inside the limit and outside, so what is the value of $n$ outside? So this doesn't make sense syntaxically

answered May 03 '15 at 12:01

Tryss

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thank you! I got confused in the constant and variable part :) – Diya May 03 '15 at 12:04

Explain why $\lim ({a_n}^k)={(\lim(a_n))}^k$ cannot be used to find limit $(1+\frac1n)^n$

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