SOLUTION: Find the equations of the horizontal and vertical asymptotes for the following. Tye none if the function does not have an asymptote. a. f(x)=2x+3/x+2 b. g(x)=5x/x^2+1

Question 124956: Find the equations of the horizontal and vertical asymptotes for the following. Tye none if the function does not have an asymptote.
a. f(x)=2x+3/x+2
b. g(x)=5x/x^2+1
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
I'll do the first one to get you started

a)

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 aysmptote, 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 aysmptote 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 aysmptote, just set the denominator equal to zero and solve for x

Set the denominator equal to zero

Subtract 2 from both sides

Combine like terms on the right side

So the vertical asymptote is

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

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