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Binomial theorem Calculator

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1

Solved example of Binomial theorem

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

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

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(1-\cos\left(3x\right)\right)}{\frac{d}{dx}\left(\tan\left(2x\right)\right)}\right)$
3

The derivative of the tangent of a function is equal to secant squared of that function times the derivative of that function, in other words, if ${f(x) = tan(x)}$, then ${f'(x) = sec^2(x)\cdot D_x(x)}$

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(1-\cos\left(3x\right)\right)}{\sec\left(2x\right)^2\frac{d}{dx}\left(2x\right)}\right)$
4

The derivative of the linear function times a constant, is equal to the constant

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

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

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

The derivative of the constant function is equal to zero

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

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

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

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 $f'(x) = -\sin(x)\cdot D_x(x)$

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(3x\right)\sin\left(3x\right)}{2\sec\left(2x\right)^2}\right)$
9

The derivative of the linear function times a constant, is equal to the constant

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

Applying the trigonometric identity: $\displaystyle\frac{1}{\sec^{n}(\theta)}=\cos^{n}(\theta)$

$\lim_{x\to0}\left(\frac{3\cos\left(2x\right)^2\sin\left(3x\right)}{2}\right)$
11

Applying an identity of double-angle cosine

$\lim_{x\to0}\left(\frac{3\left(1-2\sin\left(x\right)^2\right)^2\sin\left(3x\right)}{2}\right)$
12

The limit of the product of a function and a constant is equal to the limit of the function, times the constant: $\displaystyle \lim_{t\to 0}{\left(2t\right)}=2\cdot\lim_{t\to 0}{\left(t\right)}$

$\frac{1}{2}\lim_{x\to0}\left(3\left(1-2\sin\left(x\right)^2\right)^2\sin\left(3x\right)\right)$
13

Expand $\left(1-2\sin\left(x\right)^2\right)^2$

$\frac{1}{2}\lim_{x\to0}\left(3\sin\left(3x\right)-12\sin\left(x\right)^2\sin\left(3x\right)+12\sin\left(x\right)^{4}\sin\left(3x\right)\right)$
14

The limit of a sum of two functions is equal to the sum of the limits of each function: $\displaystyle\lim_{x\to c}(f(x)\pm g(x))=\lim_{x\to c}(f(x))\pm\lim_{x\to c}(g(x))$

$\frac{1}{2}\left(\lim_{x\to0}\left(3\sin\left(3x\right)\right)+\lim_{x\to0}\left(-12\sin\left(x\right)^2\sin\left(3x\right)\right)+\lim_{x\to0}\left(12\sin\left(x\right)^{4}\sin\left(3x\right)\right)\right)$
15

Solve the product $\frac{1}{2}\left(\lim_{x\to0}\left(-12\sin\left(x\right)^2\sin\left(3x\right)\right)+\lim_{x\to0}\left(12\sin\left(x\right)^{4}\sin\left(3x\right)\right)\right)$

$\frac{1}{2}\lim_{x\to0}\left(3\sin\left(3x\right)\right)+\frac{1}{2}\lim_{x\to0}\left(-12\sin\left(x\right)^2\sin\left(3x\right)\right)+\frac{1}{2}\lim_{x\to0}\left(12\sin\left(x\right)^{4}\sin\left(3x\right)\right)$
16

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

$\frac{1}{2}\sin\left(3\cdot 0\right)3+\frac{1}{2}\lim_{x\to0}\left(-12\sin\left(x\right)^2\sin\left(3x\right)\right)+\frac{1}{2}\lim_{x\to0}\left(12\sin\left(x\right)^{4}\sin\left(3x\right)\right)$
17

Simplifying

$0+\frac{1}{2}\lim_{x\to0}\left(-12\sin\left(x\right)^2\sin\left(3x\right)\right)+\frac{1}{2}\lim_{x\to0}\left(12\sin\left(x\right)^{4}\sin\left(3x\right)\right)$
18

$x+0=x$, where $x$ is any expression

$\frac{1}{2}\lim_{x\to0}\left(12\sin\left(x\right)^{4}\sin\left(3x\right)\right)+\frac{1}{2}\lim_{x\to0}\left(-12\sin\left(x\right)^2\sin\left(3x\right)\right)$
19

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

$\frac{1}{2}\sin\left(3\cdot 0\right)\sin\left(0\right)^{4}12+\frac{1}{2}\lim_{x\to0}\left(-12\sin\left(x\right)^2\sin\left(3x\right)\right)$
20

Simplifying

$0+\frac{1}{2}\lim_{x\to0}\left(-12\sin\left(x\right)^2\sin\left(3x\right)\right)$
21

$x+0=x$, where $x$ is any expression

$\frac{1}{2}\lim_{x\to0}\left(-12\sin\left(x\right)^2\sin\left(3x\right)\right)$
22

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

$\frac{1}{2}\sin\left(3\cdot 0\right)\sin\left(0\right)^2-12$
23

Simplifying

$0$

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