The natural logarithmic limit rule is a most useful formula for dealing the logarithmic functions in the limits. According to this limit formula, the limit of a rational expression in the following form is equal to one.
$\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\log_{e}{(1+x)}}{x}}$ $\,=\,$ $1$
Now, let us learn how to derive the natural logarithmic limit rule in calculus mathematically in three steps.
According to Taylor or Maclaurin series, the logarithm of $1+x$ to base $e$ can be expanded as an infinite series in terms of $x$ as follows.
$\ln{(1+x)}$ $\,=\,$ $x$ $-$ $\dfrac{x^2}{2}$ $+$ $\dfrac{x^3}{3}$ $-$ $\dfrac{x^4}{4}$ $+ \ldots$
Now, we can replace the logarithmic function by its equivalent infinite series in the rational function for evaluating its limit.
$\implies$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\log_{e}{(1+x)}}{x}}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{x-\dfrac{x^2}{2}+\dfrac{x^3}{3}-\dfrac{x^4}{4}+\ldots}{x}}$
There is a common factor $x$ in each term of the infinite series in the numerator. So, the common factor can be taken out from all the terms.
$= \,\,\,$ $\displaystyle \large \large \lim_{x \,\to\, 0}{\normalsize \dfrac{x\Bigg(1-\dfrac{x}{2}+\dfrac{x^2}{3}-\dfrac{x^3}{4}+\ldots \Bigg)}{x}}$
$= \,\,\,$ $\require{cancel} \displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\cancel{x}\Bigg(1-\dfrac{x}{2}+\dfrac{x^2}{3}-\dfrac{x^3}{4}+\ldots \Bigg)}{\cancel{x}}}$
$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \Bigg(1-\dfrac{x}{2}+\dfrac{x^2}{3}-\dfrac{x^3}{4}+\ldots \Bigg)}$
Now, we can evaluate the limit of the infinite series as $x$ tends to zero by the direct substitution.
$= \,\,\,$ $1-\dfrac{(0)}{2}+\dfrac{{(0)}^2}{3}-\dfrac{{(0)}^3}{4}+\ldots$
$= \,\,\,$ $1-0+0-0+\ldots$
$= \,\,\,$ $1$
Therefore, it’s proved that the limit of ratio of $\ln{(1+x)}$ to $x$ as $x$ closer to zero is equal to one.
$\therefore\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\ln{(1+x)}}{x}}$ $\,=\,$ $1$
A best free mathematics education website for students, teachers and researchers.
Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.
Learn how to solve the maths problems in different methods with understandable steps.
Copyright © 2012 - 2022 Math Doubts, All Rights Reserved