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Let $f(x, y)=\left\{\begin{array}{ll}1, & y \geq x^{4} \\ 1, & y \leq 0 \\ 0, & \text { otherwise }\end{array}\right.$Find each of the following limits, or explain that the limit does not exist.a. $\lim _{(x, y) \rightarrow(0,1)} f(x, y)$b. $\lim _{(x, y) \rightarrow(2,3)} f(x, y)$c. $\lim _{(x, y) \rightarrow(0,0)} f(x, y)$

a) $1,$ b) $0,$ c) DNE

Calculus 3

Chapter 14

Partial Derivatives

Section 2

Limits and Continuity in Higher Dimensions

Campbell University

Oregon State University

Harvey Mudd College

Idaho State University

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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$$\text { Let } f(x, y…

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$\operatorname{Let} f(x, y…

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$$\text { Let } f(x, y)=\l…

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Let $f(x, y)=\left\{\begin…

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If \lim _{x \rightarrow 0}…

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If $\lim _{x \rightarrow 0…

um All right, so they didn't function. Um, it's other water zero. And we have to inspect regions which it obtains those to balance, uh, on the plane. Why? Good evening. Call x before in sti part of the plane above the cortical equation, Michalek said. For and also the lower half of that playing. That's three wireless. We could zero. The function equals one in elsewhere. It zero uh, So if the Canada point is within the interior of one of the region's where was 10 uh, then the limit will exist and be respected people to ones you know, Uh, the issue, uh, issues problems with her when it lies on the boundary of those two regions. Wish, case. Someone will not occur because of any neighborhood in such a point on the punctual, huh? Be both equal to one and zero. Uh, so the problem reduces to considering the candidate. Points are in the interior, by the region or on the boundary. Uh, so in case A, we have the 0.1 we know is, uh, going to be because one is great and equal is strictly greater than you was squared. We know it's in the region where At his one the interior of that region. Because the Holly district s so we can say in this case gain because one is strictly greater than zero to the floor, we have that our limit will be the value of the function, that region, which is constant, a neighbor of this point in the people implement. Which is why, in this case, now, in case be in this case, we have to square in 16. And, uh, three is strictly lesson 16. Sorry to the fore and 16. And we also have that this value for why three strictly greater than zero. Um, so this tells us is that, uh, this is strictly outside of the regions which wise one, respectively. Why it leaves texted a foreign while it's equal to zero. Uh, which is the same as saying that this point wise within the interior region in which, uh, function sins about zero. And so in a neighborhood at that point, the function is constant, uh, equal to zero. And so that is the Value village at the 0.23 And finally, in Case C were considered your did in this time on the boundary because the boundary we have zero equals for why? Due to the floor wax, we also have ah zero equals zero for the wireless nickel to zero eso this point The origin lies on the boundary of, uh to curves. Why the sex before while at zero specifying the region where? Why is one, Um And since it is on the boundary of that, the other regional allies, you know, in every neighborhood of the origin, the function seems both values 10 the supplies that somebody just like exist.

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