Question 966275
Again?  This has already been discussed AND answered.  You already have this answer!



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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.


{{{(x-4)/((x^2)-4)}}}


{{{(x-4)/(x^2-4)}}}


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:
{{{(red(x)-cross(4))/(red(x^2)-cross(4))}}}
and
as x goes toward negative infinity,  {{{x/x^2}}} tends toward BUT NEVER BECOMES zero.


...  Same with x going toward positive infinity, unbound toward the right.


The horizontal asymptote is {{{highlight(y=0)}}}.