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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.

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.

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