The limit notation of a horizontal asymptote is as x --> infinity, f(x) --> # (number value); as x --> - infinity, f(x) --> #. This is read like this, "as we go to the right, the graph approaches some number value" and "as we go to the left, the graph approaches some number." So, the graph does not go up or down, rather it approaches some number value. The graph can sometimes cross through a horizontal asymptote, but only towards the middle of the graph. The graph can never cross towards the far left or far right.
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