You should endeavor to put as much of your question as possible in the form of text/MathJax. This makes the question more accessible, e.g. to those using screen readers or who have custom fonts/text size such as for dyslexia or because they have difficulty reading small text. It also makes the question easier for search engines to index. As a bonus reason, I use a dark mode extension, but your question asks me to strain my eyes staring at blazing white boxes to figure out what you're asking.

More specifically to your question, I have no idea what you are trying to communicate with the first image. As far as I can tell, you've simply added the text "$1/t,t\ne 0$ to it. Also, doesn't simply plotting $x=1/t$ as a function of $t$ not already make it graphically and intuitively obvious that $t$ approaching $0^{\pm }$ causes $x$ to approach $\pm \mathrm{\infty }$? Other properties that might be useful for the purposes of limits such as continuity and monotonicity are also evident from the graph.

@DerekElkins The first image pictures the LHS, i.e. $\underset{x\to \pm \mathrm{\infty }}{lim}f(x)$. How can I visualize the RHS, or why the LHS = RHS, from this picture alone?

@DerekElkins "doesn't simply plotting $x=1/t$ as a function of $t$ not already make it graphically and intuitively obvious that $t$ approaching $0^{\pm }$ causes $x$ to approach $\pm \mathrm{\infty }$" Perhaps...but how can I visualize this change of variable right in this image, without plotting another graph?

@DerekElkins "your question asks me to strain my eyes staring at blazing white boxes to figure out what you're asking." Apology! But I can't render pictures into text or MathJax???

I did not say that you needed to "render pictures as text/MathJax", I said to put as much of your question as possible in the form of text/MathJax. There is absolutely no reason for the second image. The relevant content of your second image is simply the equations 8 and 9 which you can easily render with MathJax. It would even be clearer (even ignoring the several issues I mentioned) as it wouldn't be cluttered with irrelevant information with the equations in weird locations.