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Show that the limits do not exist.$$\lim _{(x, y) \rightarrow(1,-1)} \frac{x y+1}{x^{2}-y^{2}}$$

does not exist.

Calculus 3

Chapter 14

Partial Derivatives

Section 2

Limits and Continuity in Higher Dimensions

Johns Hopkins University

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Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Show that the limits do no…

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Find the limit, if it exis…

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Show that the limit does n…

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they were given that the limit of X Y approaches one negative one of X Y plus one over X squared minus, twice squared. And we want to show this lemon does not exist basically one once to find limit along two different pathways and show that they're not equal. So for this one, just to be simple, I'm going to go ahead and have my auntie pathways B Along X equals run along y equals negative one. So first, to start off with him and do along why equals negative one. So when I do that, I get the limit as X approaches one of an honest replacing each Why was negative one? So I get negative X plus one over X squared minus one, and then I can factor the bottom here, since that is a difference of squares. So when I factor the bottom, I get X plus one X minus one and then on the top I want to look like the ones on the bottom here, someone affect or a negative one. When I do that, I get negative X minus one s. So then these two things cancel. So I end up with a limit as X approaches one of negative one over. It's this one which is equal to negative 1/2. And then I want to go along. X equals one the other there. So I'm given limit as y approaches. Negative one. Now I'm gonna replace every ex with one so on the top and left with y plus ones over on the bottom. I'm left with one minus y squared. So if I factor the bottom, I get it with different squares. One plus why one minus y and then 19 Cancel things out so I can cancel both of those. Some left with limit as y approaches native one of one over one minus y. And when I plug in a negative one here, end up with 1/2. And so I see there are close to being equal. However, when I go along, y equals negative one. I get a negative 1/2. So therefore, lim, it does not exist

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