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

Algebra ->  Algebra  -> Quadratic-relations-and-conic-sections -> 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      Log On

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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(21685) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first one to get you started

a)

y=%282x%2B3%29%2F%28x%2B2%29%29 Start with the given function



Looking at the numerator 2x%2B3, we can see that the degree is 1 since the highest exponent of the numerator is 1. For the denominator x%2B2, we can see that the degree is 1 since the highest exponent of the denominator is 1.


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 2x%2B3, the leading coefficient is 2

Looking at the denominator x%2B2, the leading coefficient is 1

So the horizontal aysmptote is the ratio of the leading coefficients. In other words, simply divide 2 by 1 to get %282%29%2F%281%29=2


So the horizontal asymptote is y=2





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

x%2B2=0 Set the denominator equal to zero


x=0-2Subtract 2 from both sides


x=-2 Combine like terms on the right side


So the vertical asymptote is x=-2


Notice if we graph y=%282x%2B3%29%2F%28x%2B2%29, we can visually verify our answers:

drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0Agraph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%282x%2B3%29%2F%28x%2B2%29%29%2C%0D%0Ablue%28line%28-20%2C2%2C20%2C2%29%29%2C%0D%0Agreen%28line%28-2%2C-20%2C-2%2C20%29%29%0D%0A%29 Graph of y=%282x%2B3%29%2F%28x%2B2%29%29 with the horizontal asymptote y=2 (blue line) and the vertical asymptote x=-2 (green line)