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$\displaystyle \large \lim_{x\,\to\,\infty}{\normalsize \bigg(\dfrac{1}{x}\bigg)}$ Rule


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


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,

  1. The quotient of $1$ divided by $x$ is $1$, when $x$ is equal to $1$.
  2. After $1$, the multiplicative inverse of $x$ is decreased as the value of $x$ is increased.
  3. 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.

limit 1/x rule at infinity graph

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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