why does a function have an asymptote 

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Simple, lets take a function 1/x for example. Well we have this function whose life's ambition is to be 0. We see this because 1/x gets smaller and smaller as it approaches big numbers. Can you find any value for x that would make 1/x = 0? No! Multiply each side by x! (That was an exclamation point... don't multiply by x factorial) That would be 0x = 1! That is not possible (in the sense of ordinal numbers). Yet the function gets smaller and smaller trying to reach a destination that it can never really reach... And that is the asymptote!