What I am trying to get at is the sort of initial assessment of a function, or in other words, the line of reasoning employed before any calculations, that leads to the conclusion that an asymptote of such and such a kind is present.
From what I gather, an asymptote is a value which the function never reaches, it only “approaches” it. But the connection between this idea and a mathematical expression is veiled in darkness, so likewise are the procedures—although I can perform them.
I apologize for the perhaps broad or vague question. My ignorance is such that I cannot see exactly what it is that confuses me, and so I cannot formulate a more precise question.
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You can read more [here](https://www.cuemath.com/calculus/asymptotes/), but in short, for a traditional polynomial function, horizontal or vertical asymptotes will exist if the function has a finite limit as x or y approach infinity/negative infinity, and a slant asymptote may be identified if the function is a division of two polynomials, with the numerator being one degree higher than the denominator.
All trigonometric and conic functions either have no asymptote, or else one or more that can be identified numerically.
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