What Defines a Line as an Asymptote?

[lntz]

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 [itex]f : A \longrightarrow B[/itex] be a function and [itex]A \subset R[/itex], [itex]B \subset R[/itex]. 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 [itex](x_{0}, 0)[/itex]
and we have [itex]lim_{x \longrightarrow x_{0}} |f(x)| = \infty[/itex]

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

I think I understand the first condition. As the values of [itex]x[/itex] approach some value [itex]x_{0}[/itex] the value of y tends towards infinity. i.e it tends towards a vertical straight line through [itex](x_{0}, 0)[/itex]. 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 [itex]x[/itex] tends towards a value [itex](x_{0}[/itex] 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.

Thanks for any help you can give, and sorry for my bad LaTeX limits...

Jacob.

 

[Mark44]

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.

 

[Keyn]

Hey, this is how I picture it, may help you..

Vertical asymptote:

[itex]lim_{x \longrightarrow x_{0}} |f(x)| = \infty[/itex] where [itex] x_{0}[/itex] is a critical point.Horizontal asymptote:

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

 

[platetheduke]

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.

 

[CellCoree]

Hello Jacob,

An asymptote is a line that a graph approaches but never touches. It can be vertical, horizontal, or oblique. The conditions listed in the definition you provided are specific to a function f(x) and its graph. The first condition states that if there is a vertical asymptote at x = x_{0}, then the limit of the function as x approaches x_{0} is infinity. This means that the graph of the function gets closer and closer to the vertical line x = x_{0} as x gets larger or smaller, but it never actually touches the line.

The second condition is for horizontal asymptotes. It states that if there is a horizontal asymptote at y = g(x), then the difference between the function f(x) and the linear function g(x) approaches 0 as x gets larger or smaller. This means that the graph of the function gets closer and closer to the horizontal line y = g(x) as x gets larger or smaller, but it never actually touches the line.

So, in the case of the exponential function, it has a horizontal asymptote at y = 0. This means that as x gets larger or smaller, the difference between the exponential function and the horizontal line y = 0 approaches 0. This is because the exponential function approaches 0 as x approaches infinity or negative infinity.

I hope this helps clarify the definition of an asymptote for you. If you have any further questions, please don't hesitate to ask. Keep up the good work in understanding mathematical concepts!