## 1)Discontinuity vs Hole 2)Vertical Asymptote

Complex numbers, rational functions, logarithms, sequences and series, matrix operations, etc.
Mel_3
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### 1)Discontinuity vs Hole 2)Vertical Asymptote

Helping Grandaughter with her Algerbra 2.
Been awhile since I've done this stuff
Hopefully some kind soul will _carefully_ check & correct me on the following...

Given a Rational Function Fx = Px/Qx

1 - When Qx = 0 you have a "discontinuity" (Because you can't divide by zero)

2 - To find the points of discontinuity she simply sets the denominator equal to zero and solves for the variable.

3 - You also have a vertical asymptote at these points UNLESS the numerator is also equal to zero.

4 - If both the numberator and the demoniator equal zero at the same time... for the same value of X... you have a "hole" ... .right?

5 - The point at which the "hole" occurs is actually a point of discontinuity as well... but there is no vertical asymptote... just a break in the graph... right?

Thanks for any help.

stapel_eliz
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Mel_3 wrote:Given a Rational Function Fx = Px/Qx

1 - When Qx = 0 you have a "discontinuity" (Because you can't divide by zero)

Yes. Also, when Qx = 0, you generally get a vertical asymptote on the graph.

Mel_3 wrote:2 - To find the points of discontinuity she simply sets the denominator equal to zero and solves for the variable.

Exactly.

Mel_3 wrote:3 - You also have a vertical asymptote at these points UNLESS the numerator is also equal to zero.

You'll draw the vertical asymptote, unless the numerator is zero for the same x-value. In this latter case, you draw an open dot.

Mel_3 wrote:4 - If both the numberator and the demoniator equal zero at the same time... for the same value of X... you have a "hole" ... .right?

Yes; that's what you're indicating on the graph by the open dot.

Mel_3 wrote:5 - The point at which the "hole" occurs is actually a point of discontinuity as well... but there is no vertical asymptote... just a break in the graph... right?

The "right" answer to that question will vary with the book and the instruction, but I believe that the majority would agree with that you've just presented.

Martingale
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### Re: 1)Discontinuity vs Hole 2)Vertical Asymptote

Mel_3 wrote:Helping Grandaughter with her Algerbra 2.
Been awhile since I've done this stuff
Hopefully some kind soul will _carefully_ check & correct me on the following...

Given a Rational Function Fx = Px/Qx

...
4 - If both the numberator and the demoniator equal zero at the same time... for the same value of X... you have a "hole" ... .right?

...

What if you have the function

$F(x)=\frac{x-1}{(x-1)^2}$