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I was learning a lesson in math about graphing, and I mashed a few random things together in my calculator (a TI-84 Plus CE) to create $y=\left|\left(\sqrt[3]{x-3}+9\right)^5\right|$ in a graph. Upon graphing, I saw nothing, so I used ZoomFit to find it. Upon graphing again, I saw this:

enter image description here

I would like to know why there is a gap in the line. I have never seen it before.

If you want to find it with ZoomFit then here is the Window settings.

enter image description here

If you need any more information, then just ask.

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This is more a matter of the TI-84 than math. At x=3, the line tangent to the curve is vertical. This means that as you get closer and closer to 3, the amount that y goes up for each step in x get larger and larger. On either side of the gap, the graph is indeed made up of vertical pixels. When the calculator gets to exactly x = 3, something happens with its graphing algorithm where the slope is so high that it just jumps to the next part of the graph. You should try playing around with your $\Delta X$ and TraceStep and see how they affect the gap.

You can also try Y = $\sqrt[3]{X}$ and see whether a similar thing happens at x = 0.

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Great question! This is a testament to finite precision of computational tools. The reason is because your function is not differentiable (in this case, the slope becomes infinite at X = 3). The calculator can only plot so many points, and so due to the steepness, there appears a large gap near this point.

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  • $\begingroup$ As a testament to this, it is probably possible to "move" the gap by changing the limits of the zoom in this particular calculator... $\endgroup$ – abiessu May 3 '18 at 17:20
  • $\begingroup$ My reason to not picking this is as it is believed, you cannot have a point of infinite value. $\endgroup$ – Liam Mclaughlin May 3 '18 at 20:59
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For the machine, the numbers are isolated points..

the number just after $3$ is some $3^+=3+\epsilon $ and just before $3$ is $3^-=3-\epsilon$.

so $$f (3^+)=(9+\epsilon^\frac13)^5$$ $$\approx 9^5 (1+5\frac {\epsilon^\frac13}{9})$$

and $$f (3^-)=(9-\epsilon^\frac13)^5$$ $$\approx 9^5 (1-5\frac {\epsilon^\frac13}{9})$$

the gap is $$f (3^+)-f (3^-)\approx 10.9^4.\epsilon^\frac13$$

which can be seen for some small computers.

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