回顾：函数是什么?

One situation in which we're often interested in limits is when we're working with a function over the reals.

See Expii's pages on functions and the real numbers for explanations of these individual topics.

A function over the reals is a function whose domain is $\mathbb{R}$, the set of real numbers, and whose values are all real numbers. In other words, it's a function such that we can plug in any real number, and the function always outputs a real number.

Here are a few points of clarification:

(1) The range doesn't necessarily have to be $\mathbb{R}$, but it must be a subset of $\mathbb{R}$. In other words, the function doesn't need to be able to output every real number. For example, the function $f(x) = x^{2} + 4x$ is a function over the reals:

(2) For the function $f(x) = x^{2} + 4x$, it makes sense to plug in complex numbers, so this function could also be considered a function over the complex numbers. But we can choose to think of it as a function over the reals; this means that we won't be plugging in any complex numbers.