Final Answer
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 $0^2$
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'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 $f'(x) = nx^{n-1}$
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'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 $1$
After deriving both the numerator and denominator, the limit results in
Evaluate the limit $\lim_{x\to0}\left(\frac{\cos\left(x\right)}{2}\right)$ by replacing all occurrences of $x$ by $0$
The cosine of $0$ equals $1$
Divide $1$ by $2$