Let \( f(x) \) be a differentiable function in \( [-1, \infty) \) and \( f(0)=1 \) such that \[ ....

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Let \( f(x) \) be a differentiable function in \( [-1, \infty) \) and \( f(0)=1 \) such that
\( \mathrm{P} \)
\[
\lim _{t \rightarrow x+1} \frac{t^{2} f(x+1)-(x+1)^{2} f(t)}{f(t)-f(x+1)}=1
\]

Find the value of \( \lim _{x \rightarrow 1} \frac{\ln (f(x)-\ln 2)}{x-1} \)


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