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1. what actually are asymptotes and could you give some examples
2. (Original post by Mf1999)
what actually are asymptotes and could you give some examples
Asymptotes are functions which your main function approaches as tends to a particular value or

For example; the asymptotes of are and because for those values the function is not defined. If I was to change the denominator into then I still retain the two asymptotes as above but I also have one more which is at as the function is now not defined at that point.

Another example; has asymptotes at so the graph approaches , and , and etc...

Another example; has an asymptote at 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 the function tends to 1 hence why y=1 is the asymptote.

Final example; has an asymptote of because the original function can be represented as so as you have so this is an example of a non-linear asymptote that the original graph approaches.
3. (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

RDKGames has provided some good examples and explanations above which are probably of more use

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