[tex] \lim_{x \to 1\ } \frac{x}{ x^{2} +1}= \frac{1}{1+1}= \frac{1}{2} \\ \lim_{x\to \infty} xf(x)= \lim_{x \to \infty} x* \frac{x}{ x^{2} +1} = \lim_{x \to \infty} \frac{x^2}{ x^{2} +1} = \\ = \lim_{x \to \infty} \frac{x^2}{ x^{2} (1+ \frac{1}{ x^{2} }) } =1[/tex]
