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

Formula

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

The limit of quotient of natural logarithm of sum of one and a variable by the variable as the input approaches zero is equal to one. It’s a standard result in calculus to find the limits of logarithmic functions.

Introduction

$x$ is a variable, and the natural logarithm of sum of $1$ and $x$ is written as $\ln{(1+x)}$ or $\log_{e}{(1+x)}$. The limit of ratio of $\ln{(1+x)}$ to $x$ as $x$ approaches $0$ is often appeared in calculus. Hence, this standard result is used as a limit rule in mathematics.

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

Mathematically, the limit of quotient of $\log_{e}{(1+x)}$ by $x$ as $x$ tends to $0$ is equal to $1$.

Other forms

Remember, the property of limit of logarithmic function can be written in several forms too.

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

$(2) \,\,\,$ $\displaystyle \large \lim_{z \,\to\, 0}{\normalsize \dfrac{\ln{(1+z)}}{z}}$ $\,=\, 1$

Proof

Learn how to prove the limit of quotient of $\log_{e}{(1+x)}$ by $x$ is equal to $1$ in mathematics.

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