Find the following limit.

To find the limit of the expression:

\[ 
\lim_{x \rightarrow 0} \frac{1 - \cos(x^2)}{x^2 \sin(x^2)}
\]

we can use Taylor series approximations to simplify the expressions for \(\cos(x^2)\) and \(\sin(x^2)\) near \(x = 0\). The Taylor series expansions for \(\cos(x)\) and \(\sin(x)\) are:

- \(\cos(x) \approx 1 - \frac{x^2}{2}\)
- \(\sin(x) \approx x\)

### Approximating \(\cos(x^2)\):
Applying the Taylor series expansion, we have:

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