## 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 *x*² + *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*