# Binomial theorem Calculator

## Get detailed solutions to your math problems with our Binomial theorem step by step calculator. Sharpen your math skills and learn step by step with our math solver. Check out more online calculators here.

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### Difficult Problems

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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