SOLUTION: Find the asymptotes of {{{y=(3x)/(x^2-1))}}}

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Question 140335: Find the asymptotes of y=%283x%29%2F%28x%5E2-1%29%29



Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
y=%283x%29%2F%28x%5E2-1%29%29 Start with the given function



Looking at the numerator 3x, we can see that the degree is 1 since the highest exponent of the numerator is 1. For the denominator x%5E2-1, we can see that the degree is 2 since the highest exponent of the denominator is 2.


Horizontal Asymptote:

Since the degree of the numerator (which is 1) is less than the degree of the denominator (which is 2), the horizontal asymptote is always y=0

So the horizontal asymptote is y=0



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Vertical Asymptote:
To find the vertical asymptote, just set the denominator equal to zero and solve for x

x%5E2-1=0 Set the denominator equal to zero


x%5E2=0%2B1Add 1 to both sides


x%5E2=1 Combine like terms on the right side


x=0%2B-sqrt%281%29 Take the square root of both sides


x=1 or x=-1 Simplify

So the vertical asymptotes are x=1 and x=-1

Notice if we graph y=%283x%29%2F%28x%5E2-1%29, we can visually verify our answers:

Graph of y=%283x%29%2F%28x%5E2-1%29%29 with the horizontal asymptote y=0 (blue line) and the vertical asymptotes x=1 and x=-1 (green lines)