# 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:

 x y = 2x + 1 1,5 4 1,3 3,6 1,1 3,2 1,05 3,1 1,02 3,04 1,01 3,02
 x y = 2x + 1 0,5 2 0,7 2,4 0,9 2,8 0,95 2,9 0,98 2,96 0,99 2,98 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 (x 1), 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. Next: Threshold Properties