Step-by-step Solution

Evaluate the limit of $\frac{1-2\cos\left(x\right)+\cos\left(x\right)^2}{1-\cos\left(x\right)}$ as $x$ approaches $0$

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Step-by-step explanation

Problem to solve:

$\lim_{x\to 0}\left(\frac{1-2\cos\left(x\right)+\cos\left(x\right)^2}{1-\cos\left(x\right)}\right)$
1

The trinomial $1-2\cos\left(x\right)+\cos\left(x\right)^2$ is perfect square, because it's discriminant is equal to zero

$\Delta=b^2-4ac=-2^2-4\left(1\right)\left(1\right) = 0$
2

Using the perfect square trinomial formula

$a^2+2ab+b^2=(a+b)^2,\:where\:a=\sqrt{\cos\left(x\right)^2}\:and\:b=\sqrt{1}$
3

Factoring the perfect square trinomial

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

Expand $\left(\cos\left(x\right)-1\right)^{2}$

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

Expand $\left(\cos\left(x\right)-1\right)^{2}$

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

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

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

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

$\lim_{x\to0}\left(\frac{2\left(\cos\left(x\right)-1\right)\frac{d}{dx}\left(\cos\left(x\right)-1\right)}{\frac{d}{dx}\left(1-\cos\left(x\right)\right)}\right)$
8

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

$\lim_{x\to0}\left(\frac{2\left(\cos\left(x\right)-1\right)\left(\frac{d}{dx}\left(\cos\left(x\right)\right)+\frac{d}{dx}\left(-1\right)\right)}{\frac{d}{dx}\left(1-\cos\left(x\right)\right)}\right)$
9

The derivative of the constant function ($-1$) is equal to zero

$\lim_{x\to0}\left(\frac{2\left(\cos\left(x\right)-1\right)\frac{d}{dx}\left(\cos\left(x\right)\right)}{\frac{d}{dx}\left(1-\cos\left(x\right)\right)}\right)$
10

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

$\lim_{x\to0}\left(\frac{-2\left(\cos\left(x\right)-1\right)\sin\left(x\right)}{\frac{d}{dx}\left(1-\cos\left(x\right)\right)}\right)$
11

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

$\lim_{x\to0}\left(\frac{-2\left(\cos\left(x\right)-1\right)\sin\left(x\right)}{\frac{d}{dx}\left(1\right)+\frac{d}{dx}\left(-\cos\left(x\right)\right)}\right)$
12

The derivative of the constant function ($1$) is equal to zero

$\lim_{x\to0}\left(\frac{-2\left(\cos\left(x\right)-1\right)\sin\left(x\right)}{\frac{d}{dx}\left(-\cos\left(x\right)\right)}\right)$
13

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

$\lim_{x\to0}\left(\frac{-2\left(\cos\left(x\right)-1\right)\sin\left(x\right)}{-\frac{d}{dx}\left(\cos\left(x\right)\right)}\right)$
14

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

$\lim_{x\to0}\left(\frac{-2\left(\cos\left(x\right)-1\right)\sin\left(x\right)}{\sin\left(x\right)}\right)$
15

Simplify the fraction by $\sin\left(x\right)$

$\lim_{x\to0}\left(-2\left(\cos\left(x\right)-1\right)\right)$
16

Solve the product $-2\left(\cos\left(x\right)-1\right)$

$\lim_{x\to0}\left(-2\cos\left(x\right)+2\right)$
17

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

$\lim_{x\to0}\left(-2\cos\left(x\right)\right)+\lim_{x\to0}\left(2\right)$
18

The limit of a constant is just the constant

$\lim_{x\to0}\left(-2\cos\left(x\right)\right)+2$
19

Evaluate the limit by replacing all occurrences of $x$ by $0$

$-2\cos\left(0\right)+2$
20

Simplifying

$-2+2$
21

Subtract the values $2$ and $-2$

$0$

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Problem Analysis

$\lim_{x\to 0}\left(\frac{1-2\cos\left(x\right)+\cos\left(x\right)^2}{1-\cos\left(x\right)}\right)$

Limits

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