Final Answer
0
Step-by-step Solution
Specify the solving method
1
If we directly evaluate the limit $\lim_{x\to 0}\left(\frac{x^2}{\sin\left(x\right)}\right)$ as $x$ tends to $0$, we can see that it gives us an indeterminate form
$\frac{0}{0}$
2
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
$\lim_{x\to 0}\left(\frac{\frac{d}{dx}\left(x^2\right)}{\frac{d}{dx}\left(\sin\left(x\right)\right)}\right)$
3
After deriving both the numerator and denominator, the limit results in
$\lim_{x\to0}\left(\frac{2x}{\cos\left(x\right)}\right)$
4
Evaluate the limit $\lim_{x\to0}\left(\frac{2x}{\cos\left(x\right)}\right)$ by replacing all occurrences of $x$ by $0$
$\frac{2\cdot 0}{\cos\left(0\right)}$
5
Multiply $2$ times $0$
$\frac{0}{\cos\left(0\right)}$
6
The cosine of $0$ equals $1$
$\frac{0}{1}$
7
Divide $0$ by $1$
0
Final Answer
0