#Question on approaching limits along arbitrary paths in multivariable calculus

32 messages · Page 1 of 1 (latest)

plain locust
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I have a question about the use of the substitution y = mx when proving that a limit $\lim\limits_{(x,y) \to (x_0,y_0)} f(x,y)$ does not exist. I understand that this method involves choosing a specific path to approach the point $(x_0, y_0) $, but I am struggling to understand what justifies this approach.

For example, if f(x,y) represents an ellipsoid, the path given by y = mx does not lie on the surface of the ellipsoid itself. What gives us the right to approach $(x_0, y_0) $ along a path that is not part of the function’s graph? It seems like we are imposing an external trajectory rather than following the function’s natural shape. I understand that limits in multiple variables allow for different approaches, but I am unsure why we are allowed to choose paths that are not based on the behaviour of the function itself.
What am I misunderstanding???

opal flintBOT
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raven cairnBOT
worn minnow
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The path you choose cannot be completely arbitrary. It has to go throught the point (x_0, y_0). Does that answer the question?

plain locust
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It seems to me like we don’t care about approaching the point through the graph but through any path that we can that is in the domain of f, is that right?

worn minnow
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@plain locust What do you mean by the graph? The graph of z = f(x,y) is a surface. (A 2D surface embedded into 3D space). The domain of f(x,y) is either the xy-plane or a subset of the xy-plane. As long as your path is is the domain and goes through the correct point, it should be fine.

plain locust
deft quail
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why would it

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the image of the path is on the surface, naturally

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the path itself is a curve on the xy plane that passes through the given point

worn minnow
plain locust
plain locust
worn minnow
plain locust
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We can’t arbitrarily choose any path to reach a point

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Meanwhile with two variables we can?

worn minnow
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We don't follow the graph of f. The inputs go along the x-axis. The outputs along the y. We can follow the graph as well, but it isn't the path of the input. We can do the same thing in the case z = f(x,y). Here the inputs are in the xy-plane. The graph here will be a curve embedded in 3D. We can follow this curve, but it isn't the path of the inputs in the xy-plane.

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@plain locust Think about when you first were learning limits and you "plugged in" closer and closer values. For example, if you have a limit as x->3, you plug in 3.1, 3.01, 3.001, etc. and also 2.9, 2.99, 2.999, etc. This path is NOT on the graph. of y=f(x).

plain locust
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So they have to be on the graph of f(x) don’t they?

worn minnow
plain locust
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Every value it can take

worn minnow
plain locust
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And vice versa

worn minnow
plain locust
golden dewBOT
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@plain locust

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