# Definition of an Asymptote

Hello,

i'm having some trouble understanding the definition of an asymptote, or rather the conditions that must be met in order for a line to be one.

I have;

"Let $f : A \longrightarrow B$ be a function and $A \subset R$, $B \subset R$. A straight line is called an asymptote if one of the following conditions is met;

1. The straight line is vertical (to the x-axis) and goes through a point $(x_{0}, 0)$
and we have $lim_{x \longrightarrow x_{0}} |f(x)| = \infty$

2. The straight line can be described as an affine linear function, that is as $g(x) = mx + c$ and we have either $lim_{x \longrightarrow \infty} (f(x) - g(x)) = 0$ or $lim_{x \longrightarrow - \infty} (f(x) - g(x)) = 0$"

I think I understand the first condition. As the values of $x$ approach some value $x_{0}$ the value of y tends towards infinity. i.e it tends towards a vertical straight line through $(x_{0}, 0)$. This fits the mental idea I had of an asymptote, but can it be applied to a function that has a horizontal asymptote such as the exponential function for example.

Perhaps this is where the second condition comes in, to cover those cases, but I am struggling to see what is going on...

Does it say that as $x$ tends towards a value $(x_{0}$ the y value of the functions are equal since their difference is zero?

I don't see how this covers the scenario of horizontal asymptotes unless it's ok to turn the argument around the other way.

Jacob.

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Mark44
Mentor
Horizonatal asymptotes are covered if you let m = 0 in the equation g(x) = mx + c.

BTW, I changed all of your [ tex ] tags to [ itex ] (for inline LaTeX). The [ tex ] tags render their contents on a separate line, which breaks up expressions that probably shouldn't be broken up.

Also, I find it easier to use ## in place of [ itex ] and  in place of [ tex ]. Whichever one you use, put a pair of these symbols at the front and rear of the expression you're working with.

Vertical asymptote:

$lim_{x \longrightarrow x_{0}} |f(x)| = \infty$ where $x_{0}$ is a critical point.

Horizontal asymptote:

$lim_{x \longrightarrow \infty} f(x) =$ any finite number
And you shall check +$\infty$ and -$\infty$

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A the line ##y=mx+b## is a slant asymptote for a function ##f(x)## if ##\lim_{x\to\infty}[f(x)-mx-b]=0##.
You can replace the line with another polynomial for other types of asymptotes at infinity. Vertical and horizontal asymptotes are in others' posts.