Undetermined limit

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:

  1. 0/0 = undeterminedWe do not know what result we would obtain if this operation were performed.
  2. (± ∞) / (± ∞) = undeterminedSince we do not know the value of these large numbers, we will not be able to know the exact result.
  3. 0 × ∞ = undeterminedWe do not know if zero for a number at infinity is some exact value.
  4. ∞ - ∞ = 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 indeterminate limit

We evaluate the limit at zero:

indeterminate limit1

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:

indeterminate limit2

Once simplified, we reevaluate the limit:

indeterminate limit3

Exercise 2: Get the value of the following limit indeterminate limit4

The first thing we will do is evaluate the limit:

indeterminate limit5

We find an indeterminacy, we must find a way to simplify it; in this case, we can factor the denominator of the summed term:

undetermined limit6

If we remove the common denominator we have:

indeterminate limit7

Taking common factor 2:

undetermined limit8

Once the expression has been simplified, we evaluate the limit again:

indeterminate limit9