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.
Many times we want to know how a function behaves at infinity, sometimes it is something complex, even abstract. To consider a limit of a function at infinity We have to take into account a series of rules that will help us facilitate operations:
- ∞ + ∞ = ∞, a very large number added to another, will result in an immensely large number.
- ∞ - ∞ = undeterminedSince we do not know exactly the size of the infinites, we will not be able to know the result of this subtraction, therefore it is indeterminate.
- ± ∞ ± k = ± ∞Any number added to or subtracted from an immensely large number (negative or positive) will not affect the giant number regardless of its sign.
- ± k × ± ∞ = ± ∞Starting from the rule of multiplication signs, we can say that any number, except zero, multiplied by infinity will be infinity.
- ± ∞ × ± ∞ = ± ∞Starting from the sign multiplication rule, we say that one giant number for another will be equal to another immensely large number.
- (± ∞) / (± ∞) = undeterminedSince we do not have the value of infinities precisely, this division gives us an undetermined number.
- k / (± ∞) = 0, any known number divided by infinity will be equal to zero; an analogy would be to divide a cake into a million people, we can say that each of them did not get anything.
- 0 × ± ∞ = undetermined, it is impossible to know if in the infinity the multiplication by zero is something determined, therefore we take it as indeterminate.
It is important to know that if in any problem the limit gives us infinity, that will be the result:
Example 1: Get the value of the following limit
Substitute x = ∞ at the limit:
Applying rules 4 and 5 we have:
The result of this limit is - ∞.
Now if we have a case of indeterminacy we must find a way to get a value, so be ± ∞.
Example: Get the value of the following limit
We substitute x = ∞ in the limit:
Applying rule 5,4 and 1 we have:
Starting from rule 6:
How can we resolve this limit? We will find a way that when substituting x = ∞ does not result in indeterminacy; We will simplify the expression by dividing the numerator and denominator by x²:
Since we have simplified the expression a bit, we substitute x = ∞ in the limit again:
Applying rule 5 and 7 we have: