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?

$x/x = 1$ for any nonzero real $x$. However, $\infty$ is not a real number. Now, if you evaluate a limit and find a result that gives $\infty/\infty$, we cannot determine what the limit should be without further work. After all, "plugging in" infinity to any of 
$$
\lim_{x\to\infty}\frac{x^2}{x},\qquad \lim_{x\to\infty}\frac{x}{x},\qquad \lim_{x\to\infty}\frac{\log x}{x}
$$
will give $\infty/\infty$. However, these limits evaluate respectively to $\infty$, $1$, and $0$ (showing that if you get $\infty/\infty$ when naively evaluating the limit, it could be any real number, or even infinity!). But if you are taking a limit $$\lim_{t\to\infty}\frac{X(t)}{X(t)}$$ for some expression $X(t)$ (where $X(t)\to\infty$ as $t\to\infty$), you can say that the limit is $1$ as long as $X(t)$ is defined for sufficiently large $t$, because $X(t)/X(t) = 1$ for $X(t)$ any real number besides $0$. In the limit, we only consider arbitrarily large real numbers (i.e. not actually infinity), so in taking the limit, we can reduce, and we see
$$
\lim_{t\to\infty}\frac{X(t)}{X(t)} = \lim_{t\to\infty} 1 = 1.
$$
Now, in some branches of mathematics, we can make sense of certain expressions with infinity. For example, we can make $\Bbb{R}$ or $\Bbb{C}$ compact, an important notion in topology and analysis, by adding a "point at infinity": meaning we create a new system given by $\tilde{\Bbb R} = \Bbb{R}\cup\{\infty\}$ or $\tilde{\Bbb C} = \Bbb{C}\cup\{\infty\}$, where $\infty$ is a formal symbol designed to be suggestive (notation not standard). However, in doing so, we remove nice properties of $\Bbb{R}$ and $\Bbb{C}$: namely, $\tilde{\Bbb R}$ and $\tilde{\Bbb C}$ are not fields. In these new systems, we can define many expressions involving $\infty$, such as $$\infty + \infty = \infty, \quad r/\infty = 0, \quad\textrm{and} \quad c/0 = \infty$$ (provided $r\neq\infty$, $c\neq 0$). There are expressions that we can't define, though. These include $$\infty/\infty, \quad\infty - \infty,\quad 0\cdot\infty,\quad \textrm{and}\quad 0/0:$$ which you might recognize as indeterminate forms. The reason they are indeterminate in this extended case of $\Bbb{R}$ and $\Bbb{C}$ is similar to the reason they are indeterminate in the case of normal calculus: defining them in one way would lead to contradictions. Say we defined $\infty/\infty = 1$. Then $$1 = \infty/\infty = (c/0)/\infty = c/(0\cdot\infty).$$ We would have to say that $0\cdot\infty = c$ for every nonzero $c$ in our original field, which simply does not make sense.
