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Gives a function $f(x, y, z)$ and a positive number $\epsilon .$ In each exercise, show that there exists a $\delta>0$ such that for all $(x, y, z)$.$$\sqrt{x^{2}+y^{2}+z^{2}}<\delta \Rightarrow|f(x, y, z)-f(0,0,0)|<\epsilon$$$$f(x, y, z)=\tan ^{2} x+\tan ^{2} y+\tan ^{2} z, \quad \epsilon=0.03$$

$\delta=0.1$

Calculus 3

Chapter 14

Partial Derivatives

Section 2

Limits and Continuity in Higher Dimensions

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University of Michigan - Ann Arbor

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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04:20

Gives a function $f(x, y, …

02:00

01:43

Each of Exercises $63-66$ …

the question we have, our function is equal. Detain it squared of X plus 10. It's great of live plus changes square See that our EPS honest equal to 0.3 and we want to show that there exists adult a greater than zero such that for all x y z, we have that the square root of X squared plus y squared plus Z squared lesson Delta implies of the function x y Z minus The function of the origin is less than Absalon. So really is questions asking us to find a delta where this entire thing is true. So we're given the information for the right hand side. So we're going to use information can relate it to this square root of X squared plus y squared plus c squared and find our delta from there. So again, starting with my no, my function, absolute value of it. My at the origin, however, is just equal to zero only about that using triangle inequality. I know that this is going to be less than or equal to absolute value of each of those sound. However, since both these air squared is going to be positive So that's just queen. It's equal the same thing with absolute values. Trucked. And I know that this is going to be less than my Absalon, So I'm looking for it to be less than 0.3 So now any relate this to my delta? So this expert plus y squared Rosie screwed. Well, I know that the absolute value of X is equal to the square root of X squared, just less than an equal to experience plus y squared plus z squared. That's going to same thing for we're X y and Z za Similarly pliancy. So therefore, that means that of Civilian X. Why Anzi are all going to be less than Delta. So and obviously these don't have absolute values but still just going to be either equal to this or less than so. Looking at this, I want all this add up to 0.3 I know they're being squared each time, so you kind of just think through what I want my adults to be. And I see that if I said Delta equal, hand it in verse of 0.1, then for expert plus y script was C square square root of that less than this delta. It will fly that the absolute value of function, which we really know is less than or equal to that to just showed. Well, let's give me less than my delta being there instead. And when I put intended inverse into each of these, I end up with 0.1 sequel to 0.3 which is equal to our Absalon.

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