Calculate the limit $\lim \limits_ {x \to \infty} \left(\frac{x^2+1}{x-1}\right)$
$\lim \limits_ {x \to \infty}\frac{x^2+1}{x-1}=\lim \limits_ {x \to \infty}\frac{x(x+\frac1x)}{x(1-\frac1x)}=\lim \limits_ {x \to \infty}\frac{x+\frac1x}{1-\frac1x}=\lim \limits_ {x \to \infty}\color{red}{\underbrace{\frac{\infty +0}{1-0}}_{\text{ not formal!}}}=\infty$
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.
Best Answer
I often use
$$\ldots=\lim_ {x \to \infty}\frac{x+\frac1x}{1-\frac1x}=\left(\frac{\infty+0}{1-0}\right)=\infty$$
or directly as $x\to\infty$
$$\ldots=\frac{x+\frac1x}{1-\frac1x}\to \infty$$
In any case I suggest to avoid that one
$$\ldots=\lim_ {x \to \infty}\color{red}{\underbrace{\frac{\infty +0}{1-0}}_{\text{ not formal!}}}=\ldots$$
also in a not formal answer since we are writing the values assumed by the terms under the limit.