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Gives a function $f(x, y)$ and a positive number $\epsilon$ In each exercise, show that there exists a $\delta > 0$ such that for all $(x, y)$$$\sqrt{x^{2}+y^{2}} < \delta \Rightarrow|f(x, y)-f(0,0)| < \epsilon$$$$f(x, y)=y /\left(x^{2}+1\right), \quad \epsilon=0.05$$

$\sqrt{x^{2}+y^{2}}<\delta \Rightarrow|f(x, y)-f(0,0)|<\epsilon$

Calculus 3

Chapter 14

Partial Derivatives

Section 2

Limits and Continuity in Higher Dimensions

Missouri State University

Harvey Mudd College

University of Nottingham

Boston College

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.

04:14

04:24

Gives a function $f(x, y)$…

01:06

this question or given that f of X Y is equal to y over exper plus one, The Epsilon is equal to 0.5 and we want to show that there exists. Adults are grand zero such that we're a x y. We have a following that square root of X squared plus y squared Slesin Delta implies the f of X y minus 00 is less than epsilon. So for this question were really just wanting to find this Delta that makes this whole entire statement true. So first, I'm going to start with what I know. So no ethics wise, it was a Y over expert plus one and Absalon So I'm gonna go ahead and just simplify this right hand part first kind of helped me find what my delta should be. So for that first part, Methanex, why is just why over X squared plus one? If I put in a zero for both X and why I just get zero in? So I'm headed to admit that that's less than my up salon, which is Sierra 0.5 And so this is the same thing as absolute value of y over extra plus one is less than zero point 05 That's because those bottom part is always be positive. The Sex Square is always positive. Adam one still positive side only that's the valley sign one. So now I just want to kind of work with this top part and related tio this x square root of X squared plus y squared. So first thing to know is that since absolute value, why is equal to swear e of y squared? And I know that this ware of y squared is less than equal to the square root of X squared plus y squared. Then we have that absolute value of why is Lester angle to this by the transitive property? And so since and so since we have that yeah, ands relating it back to this x squared plus one is greater than equal to zero against positive. Then we have this whole expression absolute value. Why over X squared plus one is less than or equal to this again. That's because that's the big number. This is just over one. So this is going to me obviously smaller. And so we want This should be less than 0.5 And so that implies, then. But, um, this swear ou of x squared plus y squared needs also be less than 0.5 in orderto have that train of inequalities. This has three less This one's lessner equal to that. So we need this to be strictly less than that, since it's possible that this ends up being equal. So then therefore, well, let Delta equal there a point 05 and we get, uh, X squared. Let's y squared lesson Delta, which is their 0.5 implies that massive value. Why over X squared plus one his lessons air 0.5 which is a cool absalon.

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