Final Answer
Step-by-step Solution
Specify the solving method
If we directly evaluate the limit $\lim_{x\to 0}\left(\frac{1-2\cos\left(x\right)+\cos\left(x\right)^2}{1-\cos\left(x\right)}\right)$ 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
After deriving both the numerator and denominator, the limit results in
If we directly evaluate the limit $\lim_{x\to 0}\left(\frac{2\sin\left(x\right)-\sin\left(2x\right)}{\sin\left(x\right)}\right)$ 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
After deriving both the numerator and denominator, the limit results in
Evaluate the limit $\lim_{x\to0}\left(\frac{2\cos\left(x\right)-2\cos\left(2x\right)}{\cos\left(x\right)}\right)$ by replacing all occurrences of $x$ by $0$
Multiply $2$ times $0$
The cosine of $0$ equals
The cosine of $0$ equals
The cosine of $0$ equals
Subtract the values $2$ and $-2$
Divide $0$ by $1$