SOLUTION: Find the horizontal and vertical asymptotes {{{y=(5x)/(x^2+1)}}}

You can put this solution on YOUR website!

Start with the given function



Looking at the numerator , we can see that the degree is since the highest exponent of the numerator is . For the denominator , we can see that the degree is since the highest exponent of the denominator is .


Horizontal Asymptote:

Since the degree of the numerator (which is ) is less than the degree of the denominator (which is ), the horizontal asymptote is always

So the horizontal asymptote is



--------------------------------------------------



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

Set the denominator equal to zero


Subtract 1 from both sides


Combine like terms on the right side


Take the square root of both sides

Since you cannot take the square root of a negative number, the answer is not a real number. So in this case, there are no vertical asymptotes.




Notice if we graph , we can visually verify our answers:

Graph of with the horizontal asymptote (blue line) . Notice how there are no vertical asymptotes