# Exponent properties Calculator

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### Difficult Problems

1

Example

$\lim_{x\to0}\left(\frac{\ln\left(x+1\right)}{x}\right)$
2

As the limit results in indeterminate form, we can apply L'Hôpital's rule

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(\ln\left(1+x\right)\right)}{\frac{d}{dx}\left(x\right)}\right)$
3

The derivative of the linear function is equal to $1$

$\lim_{x\to0}\left(\frac{d}{dx}\left(\ln\left(1+x\right)\right)\right)$
4

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\lim_{x\to0}\left(\frac{1}{1+x}\cdot\frac{d}{dx}\left(1+x\right)\right)$
5

The derivative of a sum of two functions is the sum of the derivatives of each function

$\lim_{x\to0}\left(\frac{1}{1+x}\left(\frac{d}{dx}\left(1\right)+\frac{d}{dx}\left(x\right)\right)\right)$
6

The derivative of the constant function is equal to zero

$\lim_{x\to0}\left(\frac{1}{1+x}\cdot\frac{d}{dx}\left(x\right)\right)$
7

The derivative of the linear function is equal to $1$

$\lim_{x\to0}\left(\frac{1}{1+x}\right)$
8

Evaluating the limit when $x$ tends to $0$

$\frac{1}{1+0}$
9

Simplifying

$1$

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