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Limits


Intuitive notion of limit

Let the function f (x) = 2x + 1. Let's give values ​​to x approaching 1 on its right (values ​​greater than 1) and on the left (values ​​less than 1) and calculate the corresponding value of y:

xy = 2x + 1
1,54
1,33,6
1,13,2
1,053,1
1,023,04
1,013,02
x y = 2x + 1
0,52
0,72,4
0,92,8
0,952,9
0,982,96
0,992,98

We note that as x approaches 1, y approaches 3, ie when x tends to 1 (x 1), y tends to 3 (y 3), ie:

We observed that when x tends to 1, y tends to 3 and the function limit is 3.

This is the study of the behavior of f (x) When x tends to 1 (x 1). Needless to say x assume the value 1. If f (x) tends to 3 (f (x) 3), we say that the limit of f (x) When x 1 is 3, although there may be cases where for x = 1 the value of f (x) is not 3. In general, we write:

if when x is approaching The (x The), f (x) approaches B (f (x)B).

How x² + x - 2 = (x - 1)(x + 2), we have:

We can note that when x approaches 1 (x1), f (x) approaches 3, although for x= 1 we have f (x) = 2. what happens is that we look for the behavior of y when x 1. And in this case y 3. Therefore, the limit of f (x) é 3.

We write:

If g: GO IR and g (x) = x + 2, g (x) = (x + 2) = 1 + 2 = 3, although g (x)f (x) in x = 1. However, both have the same limit.

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