The Formal (and Depressingly Confusing) Definition of a Limit
Let us say f(x) is a function defined over an interval, where a value [x = a] exists over the interval. The function does not have to be defined at x = a.
The limit can be expressed as:
Now, let us define ɛ (epsilon) and δ (delta) as two numbers where, when:
When these conditions can be met, that means the limit exists.
n.b. the above definition is only used in great formality; one may need to memorize it for a quiz, but it is recommended to not dwell on it. Basically, the definition essentially says that the difference between a value close to a (x) and a is not 0 and is less than a very, very small number (delta). The difference between the limit and y-value of the corresponding x should also be less than a very, very small number (epsilon).
The Informal (and Actually Applicable) Definition of a limit
The limit of a function is the y-value that the function approaches while approaching a given x-value.
Using the above expression of a limit, let’s calculate the y-values according to x-values near x=a, getting closer and closer to a, but never actually. (For example, calculate f(a-1) and f(a+1), then f(a-0.1) and f(a+0.1), f(a-0.01) and f(a+0.01) and f(a-0.001) and f(a+0.001), and so on. The y-value that these pairs of y-values approaches is the limit L. Observe the following example: