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Let \(f:[{0,1}]\longrightarrow[{0,\,1}]\) a continuous function such that \(f(0)=0\) and for every \(x,y\in[{0,1}]\) holds: \[|{f(x)-f(y)}|\geq|{x-y}|\,.\] Let, also, the sequence \(\{{x_{n}}\}_{n=1}^{\infty}\), with \(x_1\in[{0,\,1}]\) and \[x_{n+1}=f({x_{n}})\,,\;n\in\mathbb{N}\,.\] Prove that:

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Find the limit \[\displaystyle\mathop{\lim}\limits_{x\rightarrow+\infty}{\left({\frac{x^2+5x+4}{x^2-3x+7}}\right)^{x}}\,.\]