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Limits by factoring Calculator

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1

Solved example of Limits by factoring

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

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

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

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

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

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

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

The derivative of the constant function is equal to zero

$\lim_{x\to0}\left(\frac{2x}{\frac{d}{dx}\left(-\cos\left(x\right)\right)}\right)$
6

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

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

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

$\lim_{x\to0}\left(\frac{2x}{\sin\left(x\right)}\right)$
8

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

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

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

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

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 ${f'(x) = \cos(x)\cdot D_x(x)}$

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

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

$\frac{2}{\cos\left(0\right)}$
12

Simplifying

$2$