MHB Vertical and Horizontal Asymptotes

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In the inverse variation equation y = (1 / (x - 3)) - 6, the vertical asymptote occurs where the denominator equals zero, which is at x = 3. The horizontal asymptote is determined by the behavior of the function as x approaches infinity, leading to y = -6. As x increases, the value of y approaches -6, indicating that the function levels off at this line. Understanding asymptotes involves analyzing how the function behaves as the denominator changes. This foundational knowledge is essential for graphing and interpreting the function accurately.
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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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