Is the Limit of (1 + 1/b)^b as b Approaches Infinity Equal to e?

[Bacle2]

But $+\infty$ is an extended real number, and you do have

$$\lim_{x \to +\infty} f(x) = f(+\infty)$$

when $f$ is continuous at $+\infty$.

But the function is defined as a real-valued function, I believe (meaning

implicitly so). It may or not be extendable continuously to the Riemann sphere,

but AFAIK, it was defined as a real-valued functiuon of a real variable.