Does $\frac{x}{x}=1$ when $x=\infty$? This may be a dumb question:

Does $\frac{x}{x}=1$ when $x=\infty$?


I understand why $\frac{x}{x}$ is undefined when $x=0$:  This can cause errors if an equation is divided by $x$ without restrictions.
Also, $\frac{\infty}{\infty}$ is undefined.  So when I use $\frac{x}{x}=1$ to simplify an equation, can it also lead to errors because $x$ can equal infinity?
Or is $x=\infty$ meaningless?

You cannot really say $x = \infty$ because $\infty \not \in  \mathbb{R}$
What you do is, you take the limes. Limes means not, that $x=a$, but that $x$ is getting closeser and closer to $a$. For example:
$$\lim_{x\mapsto 0}\frac{1}{x}=\infty$$ because the divisor gets smaller and smaller
$$\frac{1}{2}=0.5 \\\frac{1}{1}=1 \\\frac{1}{0.5}=2\\..$$
So this is growing and growing, but $\frac{1}{0}$ is mathematically nonsense.
So it's quite simple.
$$\lim_{x\mapsto \infty}\frac{x}{x}=\lim_{x\mapsto\infty}\frac{1}{1}=1$$
In that case you simply cancel (is that the right word in english?) the $x$. That way the Limes and the $\infty$ vansishs.
