Is $\lim _{ x\to \infty }\:x\:=\:\lim _{ t\to 0}\frac{1}{t}$? Is $\lim _{ x\to \infty }\:x\:=\:\lim _{ t\to 0}\frac{1}{t}$  ?
 A: Both the limit are not defined, therefore we cannot say that they are equal (Since both tend to infinity, and two infinities need not to be equal), By the way if the limit is finite ;
$$\lim_{x \to \infty} f(x) = a=\lim_{t \to 0^{\color{red}+}} f\Big(\frac{1}{t}\Big)  $$
And ; 
$$\lim_{x \to ~-\infty} f(x) = a = \lim_{t \to 0^{\color{red}-}} f\Big(\frac{1}{t}\Big)  $$
Where , $a \in \mathbb{R}$
A: Technically, no.
But I think it is a fair allowance of notation abuse. (Provided you specify that $t > 0$.)
Technically speaking $\lim_{x\rightarrow a} f(x) = q$ does not mean that $\lim_{x\rightarrow a} f(x)$ is a number that is equal to $q$.  It means that the function $f$, and the numbers $a$ and $q$ have a special relationship and that $q$ is unique in having this relationship with $f$ and $a$[$*$].
But, because $q$ is unique it is well-defined and consistent and meaningful to say we can denote the unique $q$ so that $\lim_{x\rightarrow a} f(x) = q$ as $\lim_{x\rightarrow a} f(x) $ a number as well.
Likewise $\lim_{x\rightarrow a} f(x) = \infty$ can not be viewed as an equality statement because $\infty$ is not a valid value.  Instead it is the statement about the behavior of function $f$ with reqards to the number $a$.[$**$]  As $\lim_{x\rightarrow a} f(x) = \infty$ and $\lim_{x\rightarrow b} g(x) = \infty$ means $f$ and $g$ have the same behavior in regards to $a$ and $b$, I believe IMO that it is abusive but fair to say   $\lim_{x\rightarrow a} f(x) = \lim_{x\rightarrow b} g(x) = \infty$ .
I would say that, although technically incorrect, it is fair to state that:
$\lim_{x\rightarrow a} f(x) =\lim_{x\rightarrow b} g(x) $ 
can be taken to mean:
$f$ and $g$ have the same behavior in regards to the values $a$ and $b$ respectively.  i) either they converge to the same value ii) they both diverge "to" either the same $\pm \infty$ or iii) they both do neither i nor ii.
It's abusive notation but it is consistent, well-defined, and unambiguous and thus, imo, acceptable.
