# Limits by L'Hôpital's rule Calculator

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### Difficult Problems

1

Example

$\lim_{x\to1}\left(l\cdot\frac{\ln\left(x\right)}{x-1}\right)$
2

The limit of the product of two functions is equal to the product of the limits of each function

$\lim_{x\to1}\left(l\right)\lim_{x\to1}\left(\frac{\ln\left(x\right)}{x-1}\right)$
3

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

$\lim_{x\to1}\left(l\right)\lim_{x\to1}\left(\frac{\frac{d}{dx}\left(\ln\left(x\right)\right)}{\frac{d}{dx}\left(x-1\right)}\right)$
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 $\displaystyle f'(x)=\frac{a'}{a}$

$\lim_{x\to1}\left(l\right)\lim_{x\to1}\left(\frac{\frac{1}{x}\cdot\frac{d}{dx}\left(x\right)}{\frac{d}{dx}\left(x-1\right)}\right)$
5

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

$\frac{\frac{1}{x}}{\frac{d}{dx}\left(x-1\right)}$
6

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

$\lim_{x\to1}\left(l\right)\lim_{x\to1}\left(\frac{\frac{1}{x}}{\frac{d}{dx}\left(-1\right)+\frac{d}{dx}\left(x\right)}\right)$
7

The derivative of the constant function is equal to zero

$\lim_{x\to1}\left(l\right)\lim_{x\to1}\left(\frac{\frac{1}{x}}{\frac{d}{dx}\left(x\right)}\right)$
8

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

$\lim_{x\to1}\left(\frac{1}{x}\right)\lim_{x\to1}\left(l\right)$
9

Apply the formula: $\frac{1}{x}$$=x^{-1}$

$\lim_{x\to1}\left(x^{-1}\right)\lim_{x\to1}\left(l\right)$
10

Evaluating the limit when $x$ tends to $1$

$1^{-1}\lim_{x\to1}\left(l\right)$
11

Simplifying

$\lim_{x\to1}\left(l\right)$
12

Evaluating the limit when $x$ tends to $1$

$l$
13

Simplifying

$l$

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