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Some expressed continued confusion:
The horizontal asymptote (if the function has one) is the UNBOUNDED behavior for the function as x tends toward either negative infinity or positive infinity. Do not focus on the function crossing this line (asymptote); just look at the what the function approaches as x goes increasing positive or increasingly negative.
As x goes unbounded to the left (meaning toward negative infinity),
the component function's highest terms become increasingly more important in the ratio of their values:
and
as x goes toward negative infinity, tends toward BUT NEVER BECOMES zero.
... Same with x going toward positive infinity, unbound toward the right.
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