** Final Answer

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** Step-by-step Solution **

Problem to solve:

** Specify the solving method

We could not solve this problem by using the method: **Limits by rationalizing**

Plug in the value $0$ into the limit

The cosine of $0$ equals $1$

Subtract the values $1$ and $-1$

Calculate the power $0^2$

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If we directly evaluate the limit $\lim_{x\to 0}\left(\frac{1-\cos\left(x\right)}{x^2}\right)$ as $x$ tends to $0$, we can see that it gives us an indeterminate form

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We can solve this limit by applying L'H么pital's rule, which consists of calculating the derivative of both the numerator and the denominator separately

Find the derivative of the numerator

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

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

The derivative of a function multiplied by a constant ($-1$) is equal to the constant times the derivative of the function

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

Any expression multiplied by $1$ is equal to itself

Find the derivative of the denominator

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

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After deriving both the numerator and denominator, the limit results in

Plug in the value $0$ into the limit

The sine of $0$ equals $0$

Multiply $2$ times $0$

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If we directly evaluate the limit $\lim_{x\to 0}\left(\frac{\sin\left(x\right)}{2x}\right)$ as $x$ tends to $0$, we can see that it gives us an indeterminate form

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We can solve this limit by applying L'H么pital's rule, which consists of calculating the derivative of both the numerator and the denominator separately

Find the derivative of the numerator

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

Find the derivative of the denominator

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

The derivative of the linear function is equal to $1$

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After deriving both the numerator and denominator, the limit results in

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Evaluate the limit $\lim_{x\to0}\left(\frac{\cos\left(x\right)}{2}\right)$ by replacing all occurrences of $x$ by $0$

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The cosine of $0$ equals $1$

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Divide $1$ by $2$

** Final Answer

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