Vertical and Horizontal Asymptotes

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SUMMARY

The discussion focuses on identifying vertical and horizontal asymptotes in the inverse variation equation y = (1 / (x - 3)) - 6. The vertical asymptote is determined by the value that makes the denominator zero, which is x = 3. The horizontal asymptote is found by analyzing the behavior of the function as x approaches infinity, resulting in y = -6. These conclusions clarify the nature of asymptotes in rational functions.

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In an inverse variation equation, what are the asymptotes and how do you find them? For example,
I was given the equation: y= [1 \ (x - 3)] - 6 and asked to find the vertical and horizontal asymptote.
I don't really understand what they are and why y= -6 and x=3. Thanks for any help!
 
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Hello and welcome to MHB, mathewslauren! (Wave)

We are given:

$$y=\frac{1}{x-3}-6$$

Now, before we discuss asymptotes, think about if you have a fraction, and you hold the numerator constant, and let the denominator vary. What happens to the value of the fraction if the denominator get larger and larger, without bound...where is the value of the fraction itself headed...and likewise, what if we let the denominator get closer and closer to zero...what happens to the value of the fraction then?
 
The fraction would get smaller as the denominator increases, and larger as it decreases.
 
mathewslauren said:
The fraction would get smaller as the denominator increases, and larger as it decreases.

Well, that's true, but can you be more specific?
 

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