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> Calculate the limit $\lim \limits\_ {x \to \infty} \left(\frac{x^2+1}{x-1}\right)$
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$\lim \limits\_ {x \to \infty}\frac{x^2+1}{x-1}=\lim \limits\_ {x \to \infty}\frac{x(x+\frac1x)}{x(1-\frac1x)}=\lim \limits\_ {x \to \infty}\frac{x+\frac1x}{1-\frac1x}=\lim \limits\_ {x \to \infty}\color{red}{\underbrace{\frac{\infty +0}{1-0}}\_{\text{ not formal!}}}=\infty$

How do I express the marked part in a formal way? I know that adding something to infinity is wrong, because $\infty$ is not a number. Unfortunately, I don't have any ideas to correct this.