Question 194181
Graph and show all asymptotes and intercepts. 
a. x/(x^2-4)
Vertical asymptotes at x= -2 and at x = 2
Horizontal at y = 0x^2/x^2 =  0/1 = 0
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Comment: Horizontal asymptotes describe the behavior of the graph
when "x" becomes very large, positively or negatively.
In your problem x^2 will overwhelm all other terms.
You have no x^2 terms in the numerator so you 
get y = 0x^2/1x^2 = 0/1 = 0 as the horizontal asymptote.
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{{{graph(400,300,-10,10,-10,10,x/(x^2-4))}}}
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b. 3/(x^2-1)
Vertical at x = -1 and at x = 1
Horizontal at y = 0
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Comment: Similarly, x^2 will overwhelm the fraction and approach zero
y = 0x^2/1x^2 = 0/1 = 0
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{{{graph(400,300,-10,10,-10,10,3/(x^2-1))}}}
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c. 2x+1/x+3 
Vertical at x = -3
Horizontal at y = 2x/x = 2/1 = 2
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The highest power of "x" is x
You have two in the numerator and one in the denominator.
You get y = 2x/x = 2 as the horizontal asymptote.
{{{graph(400,300,-10,10,-10,10,(2x+1)/(x+3))}}}
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You should be able to see all these asymptotes in the graphs.
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Cheers,
Stan H.