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Mf1999

what actually are asymptotes and could you give some examples

Reply 1

RDKGames

Study Forum Helper

Original post by Mf1999

what actually are asymptotes and could you give some examples

Asymptotes are functions which your main function approaches as

x

tends to a particular value or

\pm \infty

For example; the asymptotes of

y=\frac{1}{x+2}

are

y=0

and

x=-2

because for those values the function is not defined. If I was to change the denominator into

y_1=\frac{1}{(x+2)(x-1)}

then I still retain the two asymptotes as above but I also have one more which is at

x=1

as the function is now not defined at that point.

Another example;

y=\tan(x)

has asymptotes at

x=\frac{\pi}{2} + \pi n, \forall n \in \mathbb{N}

so the graph approaches

x=\frac{\pi}{2}

, and

x=\frac{3\pi}{2}

, and

\frac{-\pi}{2}

etc...

Another example;

y=\frac{x^2+2x+1}{x^2-5x+8}

has an asymptote at

y=1

but this is one of the scenarios where the curve DOES intersect the asymptote (at 1,1) unlike most others which are more common. The function merely has to approach the asymotote, not necessarily avoid intersecting it. So as

x\rightarrow \pm \infty

the function tends to 1 hence why y=1 is the asymptote.

Final example;

y=\frac{x^3+2x^2+7x+1}{x+2}

has an asymptote of

y=x^2+7

because the original function can be represented as

y=x^2+7-\frac{13}{x+2}

so as

x \rightarrow \pm \infty

you have

y\rightarrow x^2+7

so this is an example of a non-linear asymptote that the original graph approaches.

(edited 7 years ago)

Reply 2

KaylaB

Original post by Mf1999

what actually are asymptotes and could you give some examples

Asymptotes are points on a graph which a line doesn't usually meet as the values are undefined, and instead gets closer and closer to that and carries on infinitely.

It's usually shown graphically with dashed lines like this