Is it trivial to say $\mathop {\lim }\limits_{n \to \infty } {(1 + {k \over n})^n} = e^{k}$ Is it trivial to say 
$$\mathop {\lim }\limits_{n \to \infty } {(1 + {k \over n})^n} = e^{k},$$

considering the fact that we know
$$\mathop {\lim }\limits_{n \to \infty } {(1 + {1 \over n})^n} = e?$$
Since $\frac kn = \frac 1 {\frac nk}$ and $n = \frac nk \cdot k$ we have
$$\lim_{n\to \infty} (1+\frac kn)^n = \lim_{n\to\infty} \left( (1 + \frac{1}{\frac nk})^{\frac nk} \right)^k = \left ( \lim_{u\to\infty} (1+\frac 1u)^u \right)^k = e^k$$
If $k&lt;0$, you have
$$(1+\frac 1u)^{-u} = \left((1+\frac 1u)^u\right)^{-1} \to e^{-1} \qquad (u\to\infty)$$
By $\displaystyle\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = e,n\in\mathbb{N}$, we have
$$\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x = e,\quad x\in \mathbb{R}$$