limit $\underbrace{\lim_{n\to\infty}\frac1n+\lim_{n\to\infty}\frac1n+\cdots+\lim_{n\to\infty}\frac1n}_{n\text{ times}}$

$$\lim_{n\to\infty}\frac nn \neq \lim_{n\to\infty}n\cdot\lim_{n\to\infty}\frac1n$$

The fourth equality $$\lim_{n\to\infty} \left(n\left(\frac1n\right)\right) =n\left(\lim_{n\to\infty}\frac1n\right)\tag{*}$$ is incorrect:

• the multiple rule $$\lim_{n\to\infty} kf(n)=k\lim_{n\to\infty} f(n) \quad\text{for }k\in\mathbb R$$ is inapplicable because the multiple $k$ does not depend on $n$ (so, $k$ cannot be $n$);
• the product rule $$\lim_{n\to\infty} g(n)f(n)=\lim_{n\to\infty} g(n)\lim_{n\to\infty} f(n)$$ is also inapplicable because all the limits must exist (so, $g(n)$ cannot be $n,$ and $\displaystyle\lim_{n\to\infty}n\ne n).$

Notice also that $(*)$ is equivalent to $$\lim_{m\to\infty} \left(m\left(\frac1m\right)\right) =n\left(\lim_{m\to\infty}\frac1m\right),$$ and that $$\exists n\;\lim_{m\to\infty} \left(m\left(\frac1m\right)\right) =n\left(\lim_{m\to\infty}\frac1m\right)$$ is a false statement.