$\displaystyle \large \lim_{x\,\to\,\infty}{\normalsize \dfrac{1}{x}}$ $\,=\,$ $0$

Let a variable is denoted by a literal $x$ and it represents every positive real number, except zero ($x \ne 0$).

The limit of the reciprocal of a variable $x$ is zero as the value of $x$ approaches infinity. It is called the reciprocal limit rule as $x$ tends to infinity. Now, let’s understand how it is possible.

If $x \,=\, 100$, then $\dfrac{1}{x} \,=\, \dfrac{1}{100} \,=\, 0.01$

Now, consider the positive real numbers from one and evaluate their reciprocals to understand the functionality of one divided by $x$.

$x$ | $\dfrac{1}{x}$ |
---|---|

$1$ | $1$ |

$2$ | $0.5$ |

$3$ | $0.333333333$ |

$4$ | $0.25$ |

$5$ | $0.2$ |

$6$ | $0.166666667$ |

$7$ | $0.142857143$ |

$8$ | $0.125$ |

$9$ | $0.111111111$ |

$x$ | $\dfrac{1}{x}$ |
---|---|

$10$ | $0.1$ |

$100$ | $0.01$ |

$1000$ | $0.001$ |

$10000$ | $0.0001$ |

$100000$ | $0.00001$ |

$1000000$ | $0.000001$ |

$10000000$ | $0.0000001$ |

$100000000$ | $0.00000001$ |

$1000000000$ | $0.000000001$ |

According to the above two tables,

- The quotient of $1$ divided by $x$ is $1$, when $x$ is equal to $1$.
- After $1$, the multiplicative inverse of $x$ is decreased as the value of $x$ is increased.
- After $1$, the reciprocal of $x$ is closer to zero as the value of $x$ is increased.

The functionality of the reciprocal of $x$ can be understood clearly by drawing a graph between the positive real numbers and their reciprocal values. In this graph, the horizontal $x$-axis represents the values of $x$ and the vertical $y$-axis denotes the corresponding reciprocal values.

The graphical representation proves that the value of the reciprocal of $x$ decreases and closely reaches zero as the value of $x$ increases. Therefore, the limit of $1$ divided by $x$ is equal to $0$, as the value of $x$ approaches infinity and it is expressed in the following mathematical form in calculus.

$\,\,\,\therefore\,\,\,\,\,\,$ $\displaystyle \large \lim_{x\,\to\,+\infty}{\normalsize \dfrac{1}{x}}$ $\,=\,$ $0$

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