Deducing the presence of asymptotes in a function.

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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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Usually you look for singularities, or places where the function either goes to infinity or infinitely small.
A good example is if you see something like 1/x , as x gets closer and closer to zero, the value of 1/x just keeps getting bigger and bigger, approaching infinity.

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.

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.