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

Email subscription
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.