One of the most important limits in calculus is

$$\bbox[#F2F2F2,5px,border:2px solid black] {\lim_{x\to0}\left(1+{x}\right)^{1/x}=e}$$

where $$e$$ is an irrational number (like $$\pi$$) and is approximately 2.718281828.

We are not going to prove that such a limit exists, but we will content ourselves by plotting $$y=(1+x)^{1/x}$$ (Figure 1), and show graphically that as $$x\approx0$$, the function $$y=(1+x)^{1/x}$$ takes on values in the near neighborhood of $$2.718\cdots$$, and therefore $$e\approx2.7182$$.

As $$x$$ approaches zero from the left, $$y$$ decreases and approaches $$e$$ as a limit. As $$x$$ approaches zero from the right, $$y$$ increases and also approaches $$e$$ as a limit.

As $$x\to\infty$$, $$y$$ approaches the limit 1, and as $$x\to-1$$ from the right, $$y$$ increases indefinitely (see Table 1).

Tabel 1
$$x$$ $$y$$ $$x$$ $$y$$
$$100$$ $$1.04723$$
$$10$$ $$1.27098$$ $$-0.99$$ $$104.762$$
$$1$$ $$2.0000$$ $$-0.9$$ $$12.9155$$
$$0.5$$ $$2.2500$$ $$-0.5$$ $$4.0000$$
$$0.1$$ $$2.59374$$ $$-0.1$$ $$2.86797$$
$$0.01$$ $$2.70481$$ $$-0.01$$ $$2.732$$
$$0.001$$ $$2.71692$$ $$-0.001$$ $$2.71964$$
$$0.0001$$ $$2.71815$$ $$-0.0001$$ $$2.71842$$

Letâ€™s replace $$1/x$$ by $$u$$. So when $$x\to0$$, we have $$u\to\infty$$. Thus

$\lim_{x\to0}(1+x)^{1/x}=\lim_{u\to\infty}\left(1+\frac{1}{u}\right)^{u}=e.$

In other words:

$$\bbox[#F2F2F2,5px,border:2px solid black] {\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}=e}$$