## Final Answer

## Step-by-step explanation

Problem to solve:

Choose the solving method

Plug in the value $0$ into the limit

The cosine of $0$ equals $1$

Multiply $-1$ times $1$

Subtract the values $1$ and $-1$

Calculate the power $0^2$

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

We can solve this limit by applying L'Hôpital'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 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 $f'(x) = -\sin(x)\cdot D_x(x)$

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 $f'(x) = nx^{n-1}$

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 $0$

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

We can solve this limit by applying L'Hôpital'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

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

The limit of the product of a function and a constant is equal to the limit of the function, times the constant. Example: $\displaystyle\lim_{t\to 0}{\left(\frac{t}{2}\right)}=\lim_{t\to 0}{\left(\frac{1}{2}t\right)}=\frac{1}{2}\cdot\lim_{t\to 0}{\left(t\right)}$

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

The cosine of $0$ equals $1$

Simplifying, we get

Multiply $\frac{1}{2}$ times $1$