The general expression of a limit 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 and L is the result of the limit.
In the study of functions there are points where they have a different or abstract behavior, but at the limit of these points we find things that we do not know how to interpret and we call them indeterminacies. Let's see what we can find at the limit of a function:
- 0/0 = undeterminedWe do not know what result we would obtain if this operation were performed.
- (± ∞) / (± ∞) = undeterminedSince we do not know the value of these large numbers, we will not be able to know the exact result.
- 0 × ∞ = undeterminedWe do not know if zero for a number at infinity is some exact value.
- ∞ - ∞ = undetermined, we are not sure of the result of these infinite numbers.
Let's see some examples:
Exercise 1: Get the value of the following limit
We evaluate the limit at zero:
We find an indeterminacy. To obtain the value of the limit we must find a way to simplify it, in this case, we can make a change of variable, x = √x²:
Once simplified, we reevaluate the limit:
Exercise 2: Get the value of the following limit
The first thing we will do is evaluate the limit:
We find an indeterminacy, we must find a way to simplify it; in this case, we can factor the denominator of the summed term:
If we remove the common denominator we have:
Taking common factor 2:
Once the expression has been simplified, we evaluate the limit again: