Find the following limit

To find the limit of \(\frac{1 - \cos(x^2)}{x^2 \sin(x^2)}\) as \(x\) approaches 0, we'll use some trigonometric identities and properties of limits.

First, notice that as \(x\) approaches 0, both \(x^2\) and \(\sin(x^2)\) approach 0. Additionally, we know that \(\lim_{x \to 0} \frac{\sin(x)}{x} = 1\). We can use these facts to simplify the expression.

First, let's rewrite \(\frac{1 - \cos(x^2)}{x^2 \sin(x^2)}\) by multiplying the numerator and denominator by \(\frac{1 + \cos(x^2)}{1 + \cos(x^2)}\), which is essentially multiplying by 1:

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