One of the basic analyzes for a function is to study its continuity and the values in which it possibly does not exist. Therefore, studying the function in reduced environments of these values and observing the behavior of itself, is what we call limits of a function.
The symbology that we will use to study the limits of a function approaching some specific value is as follows:
Where lim is the abbreviated way of writing limit, x → a is read "when x tends to the value a in the function", that is, when the variable x takes values very close to the value a.
Example: Limit of the function f (x) = x + 1.
We will first analyze the domain of this function. We can quickly observe that this function has no restriction, for any real value of the variable x there will be a value f (x), therefore, its domain is all real numbers:
Now let's analyze the limit when x → 1:
In this case, since we do not have a restriction in the domain, we substitute the value to which the variable x tends in the function:
It makes sense that as x → 1 your image on the y axis is 2.
These are some properties of limits or as they are also known: limit algebra:
The limit of the sum of functions will be equal to the sum of the limit of each function. Example:
This property is true for the four basic operations:
The limit of a function raised to another function will be equal to the limit of the base function raised to the limit of the exponent function. Example:
Applying the properties let's solve the following limit: