SOLUTION: i need to find the asymptotes of the rational function of 6x^2/2x^2+5x+2

Question 145529: i need to find the asymptotes of the rational function of
6x^2/2x^2+5x+2
Answer by jim_thompson5910(35256) (Show Source): 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 and the denominator are the same, we can find the horizontal asymptote using this procedure:

To find the horizontal asymptote, first we need to find the leading coefficients of the numerator and the denominator.

Looking at the numerator , the leading coefficient is

Looking at the denominator , the leading coefficient is

So the horizontal asymptote is the ratio of the leading coefficients. In other words, simply divide by to get

So the horizontal asymptote is

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

Set the denominator equal to zero

Now let's use the quadratic formula to solve for x. If you need help with the quadratic formula, check out this solver.

After using the quadratic formula, we get the solutions
or

So this means the vertical asymptotes are or

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

Graph of with the horizontal asymptote (blue line) and the vertical asymptotes and (green lines)

note: for some reason the graph didn't format correctly, but you get the idea. This should look better on a graphing calculator.

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