Boundary evaluation

The general expression of a limit is as follows:

limits1

Where lim is the abbreviated way of writing limit, f (x) is the function under study and x → a reads "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 general, when we evaluate a limit we will find indeterminacies or values that we do not know in mathematics, however, thanks to algebra we will be able to "eliminate" these indeterminacies and thus obtain results for them. Let's see some examples:

Exercise 1: Calculate the limit  limit.

The first thing we will do is replace the value that the x tends to:

limit1

We have an indeterminacy, because in mathematics this number has no real meaning; How can we eliminate it?

Factor the limit numerator:

limit2

Now if we can evaluate when x = - 1:

limit3

So:

limit4

Exercise 2: Calculate the limit limit5

The first thing we will do is replace the value that the t tends to:

limit6

We have an indeterminacy. We are going to simplify the expression of the limit, we will start with factoring the denominator:

limit7

Now we evaluate when t = - 2:

limit8

So:

limit9

Exercise 3: Calculate the limit limit10.

The first thing we will do is replace the value that the x tends to:

limit11

We have an indeterminacy. By simplifying:

limit12

Now when evaluating x = 0:

limit13

So:

limit14