June 26, 2019

### Limit of a Function at a Point

**The limit of the function, f(x), at point, x _{0}**,

is essentially the value of

**y**when

**x**approaches

**x**.

_{0}Take for example, the function f(x) = x² at the point x_{0} = 2.

x | f(x) |
---|---|

1,9 | 3,61 |

1,99 | 3,9601 |

1,999 | 3,996001 |

... | ... |

↓ | ↓ |

2 | 4 |

x | f(x) |
---|---|

2,1 | 4.41 |

2,01 | 4,0401 |

2,001 | 4,004001 |

... | ... |

↓ | ↓ |

2 | 4 |

When x becomes closer to 2 from the left and right side the value of the function will approach 4.

**It is said that the limit of the function, f(x) , as x tends to x _{0}, is L. If a real positive number is set, ε, greater than zero, there will be a positive number, δ, depending on ε , for all the values of x that differ from x_{0} that fulfill the condition |x - x_{0}| < δ , and holds that |f(x) - L| <ε .**

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