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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Functions typically hit an asymptote when the denominator of a fraction approaches zero, so when hunting for asymptotic behavior you’ll want to find the zeroes.
For example, f(x) = 8x/(x+2)
The denominator has a zero at x=-2, so the function is undefined at that value and you can further investigate whether there’s an asymptote there or not.
It does have an asymptote here because the value at -1.9999999 and the value at -2.00000001 aren’t close, it launches off to positive infinity from one direction and negative infinity from the other.
This is the general concept of “limits” – the function is undefined at -2 but it does have a value very near -2, and that value is large.
Asymptotes exist at these undefined points with infinitely large limits.
Do note that an undefined point isn’t *always* an asymptote though. Some functions have factored terms that cancel out so they simply don’t exist at a certain value but are otherwise a smooth curve.
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