Math Doubts

Natural logarithmic Limit rule


$\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\ln{(1+x)}}{x}}$ $\,=\,$ $1$

The limit of the quotient of natural logarithm of one plus a variable by the variable as the input approaches zero is equal to one. It is called the natural logarithmic limit rule.


When the input of a function is denoted by a variable $x$, the natural logarithm of $1$ plus $x$ is written in the following two mathematical forms.

$(1).\,\,\,$ $\log_{e}{(1+x)}$

$(2).\,\,\,$ $\ln{(1+x)}$

This type of logarithmic functions appear in rational form with a variable as follows in calculus.


In some cases, we have to calculate the limit of above form rational function as the input closes to zero. It is mathematically expressed in the following mathematical form in calculus.

$\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\ln{(1+x)}}{x}}$

Actually, the limit of this type of rational function is equal to one as the input of the function tends to zero.

$\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\ln{(1+x)}}{x}}$ $\,=\,$ $1$

This standard result is used as a formula while dealing the logarithmic functions in limits.

Other forms

The natural logarithmic limit rule can be expressed in terms of any variable but it should be in the same form. Hence, the logarithmic limit rule in terms of natural logarithms can be written in the following forms too.

$(1).\,\,\,$ $\displaystyle \large \lim_{m \,\to\, 0}{\normalsize \dfrac{\ln{(1+m)}}{m}}$ $\,=\,$ $1$

$(2).\,\,\,$ $\displaystyle \large \lim_{t \,\to\, 0}{\normalsize \dfrac{\log_{e}{(1+t)}}{t}}$ $\,=\,$ $1$

$(3).\,\,\,$ $\displaystyle \large \lim_{y \,\to\, 0}{\normalsize \dfrac{\ln{(1+y)}}{y}}$ $\,=\,$ $1$

In this way, you can write the natural logarithmic limit rule in the form of any variable.


Learn how to prove the natural logarithmic limit rule in calculus from the fundamentals of mathematics.

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