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Limits by L'Hôpital's rule Calculator

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1

Example

$\lim_{x\to0}\left(\frac{x^3\cot\left(x\right)}{1-\cos\left(x\right)}\right)$
2

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

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(x^3\cot\left(x\right)\right)}{\frac{d}{dx}\left(1-\cos\left(x\right)\right)}\right)$
3

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x^3$ and $g=\cot\left(x\right)$

$\lim_{x\to0}\left(\frac{x^3\frac{d}{dx}\left(\cot\left(x\right)\right)+\cot\left(x\right)\frac{d}{dx}\left(x^3\right)}{\frac{d}{dx}\left(1-\cos\left(x\right)\right)}\right)$
4

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\lim_{x\to0}\left(\frac{x^3\frac{d}{dx}\left(\cot\left(x\right)\right)+3x^{2}\cot\left(x\right)}{\frac{d}{dx}\left(1-\cos\left(x\right)\right)}\right)$
5

Taking the derivative of cotangent

$\lim_{x\to0}\left(\frac{3x^{2}\cot\left(x\right)-x^3\csc\left(x\right)^2}{\frac{d}{dx}\left(1-\cos\left(x\right)\right)}\right)$
6

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

$\lim_{x\to0}\left(\frac{3x^{2}\cot\left(x\right)-x^3\csc\left(x\right)^2}{\frac{d}{dx}\left(-\cos\left(x\right)\right)+\frac{d}{dx}\left(1\right)}\right)$
7

The derivative of the constant function is equal to zero

$\lim_{x\to0}\left(\frac{3x^{2}\cot\left(x\right)-x^3\csc\left(x\right)^2}{\frac{d}{dx}\left(-\cos\left(x\right)\right)}\right)$
8

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\lim_{x\to0}\left(\frac{3x^{2}\cot\left(x\right)-x^3\csc\left(x\right)^2}{-\frac{d}{dx}\left(\cos\left(x\right)\right)}\right)$
9

The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if $f(x) = \cos(x)$, then $f'(x) = -\sin(x)\cdot D_x(x)$

$\lim_{x\to0}\left(\frac{3x^{2}\cot\left(x\right)-x^3\csc\left(x\right)^2}{\sin\left(x\right)}\right)$
10

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

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(3x^{2}\cot\left(x\right)-x^3\csc\left(x\right)^2\right)}{\frac{d}{dx}\left(\sin\left(x\right)\right)}\right)$
11

The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(3x^{2}\cot\left(x\right)-x^3\csc\left(x\right)^2\right)}{\cos\left(x\right)}\right)$
12

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

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(-x^3\csc\left(x\right)^2\right)+\frac{d}{dx}\left(3x^{2}\cot\left(x\right)\right)}{\cos\left(x\right)}\right)$
13

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\lim_{x\to0}\left(\frac{3\frac{d}{dx}\left(x^{2}\cot\left(x\right)\right)-\frac{d}{dx}\left(x^3\csc\left(x\right)^2\right)}{\cos\left(x\right)}\right)$
14

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\csc\left(x\right)^2$ and $g=x^3$

$\lim_{x\to0}\left(\frac{3\frac{d}{dx}\left(x^{2}\cot\left(x\right)\right)-\left(\csc\left(x\right)^2\frac{d}{dx}\left(x^3\right)+x^3\frac{d}{dx}\left(\csc\left(x\right)^2\right)\right)}{\cos\left(x\right)}\right)$
15

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x^{2}$ and $g=\cot\left(x\right)$

$\lim_{x\to0}\left(\frac{3\left(x^{2}\cdot\frac{d}{dx}\left(\cot\left(x\right)\right)+\cot\left(x\right)\frac{d}{dx}\left(x^{2}\right)\right)-\left(\csc\left(x\right)^2\frac{d}{dx}\left(x^3\right)+x^3\frac{d}{dx}\left(\csc\left(x\right)^2\right)\right)}{\cos\left(x\right)}\right)$
16

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\lim_{x\to0}\left(\frac{3\left(x^{2}\cdot\frac{d}{dx}\left(\cot\left(x\right)\right)+2x\cot\left(x\right)\right)-\left(3\csc\left(x\right)^2x^{2}+x^3\frac{d}{dx}\left(\csc\left(x\right)^2\right)\right)}{\cos\left(x\right)}\right)$
17

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\lim_{x\to0}\left(\frac{3\left(x^{2}\cdot\frac{d}{dx}\left(\cot\left(x\right)\right)+2x\cot\left(x\right)\right)-\left(3\csc\left(x\right)^2x^{2}+2x^3\frac{d}{dx}\left(\csc\left(x\right)\right)\csc\left(x\right)\right)}{\cos\left(x\right)}\right)$
18

Taking the derivative of cotangent

$\lim_{x\to0}\left(\frac{3\left(2x\cot\left(x\right)-x^{2}\csc\left(x\right)^2\right)-\left(3\csc\left(x\right)^2x^{2}+2x^3\frac{d}{dx}\left(\csc\left(x\right)\right)\csc\left(x\right)\right)}{\cos\left(x\right)}\right)$
19

Taking the derivative of cosecant

$\lim_{x\to0}\left(\frac{3\left(2x\cot\left(x\right)-x^{2}\csc\left(x\right)^2\right)-\left(3\csc\left(x\right)^2x^{2}-2x^3\csc\left(x\right)\cot\left(x\right)\csc\left(x\right)\right)}{\cos\left(x\right)}\right)$
20

When multiplying exponents with same base you can add the exponents

$\lim_{x\to0}\left(\frac{3\left(2x\cot\left(x\right)-x^{2}\csc\left(x\right)^2\right)-\left(3\csc\left(x\right)^2x^{2}-2x^3\csc\left(x\right)^2\cot\left(x\right)\right)}{\cos\left(x\right)}\right)$
21

Factoring by $\csc\left(x\right)^2$

$\lim_{x\to0}\left(\frac{3\left(2x\cot\left(x\right)-x^{2}\csc\left(x\right)^2\right)-\csc\left(x\right)^2\left(3x^{2}-2x^3\cot\left(x\right)\right)}{\cos\left(x\right)}\right)$
22

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

$\frac{\left(0^{2}\cdot \csc\left(0\right)^2\left(-1\right)+\cot\left(0\right)\cdot 0\cdot 2\right)\cdot 3-1\cdot \left(0^{2}\cdot 3+0^3\cdot \cot\left(0\right)\left(-2\right)\right)\cdot \csc\left(0\right)^2}{\cos\left(0\right)}$
23

Simplifying

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