Answered: Let f(x,y,z)=x^2+4y^2+2z^2. Does the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let f(x,y,z)=x^2+4y^2+2z^2. Does the limit lim_(0,0,0)f(x,y,z) exist? If so, use generalized to 3 dimension to show that this is the limit.

Expert Solution

Step 1

Given:-

Let f(x,y,z)= $x^{2} + 4 y^{2} + 2 z^{2}$ . Does the limit lim_(0,0,0)f(x,y,z) exist? If so, use generalized to 3 dimension to show that this is the limit.

Step 2

Solution:-

f(x,y,z) = $x^{2} + 4 y^{2} + 2 z^{2}$ .

$\lim_{(x, y, z) \to (0, 0, 0)} f (x, y, z)$

The given function is a polynomial function which is continuous everywhere so, this function is also continuous at (0,0,0)

$\lim_{(x, y, z) \to (0, 0, 0)} f (x, y, z)$ = f (0,0,0) = 0

$|f (x, y, z - 0)| = |x^{2} + 4 y^{2} + 2 z^{2}|$

= $x^{2} + 4 y^{2} + 2 z^{2}$ $\leq$ $x^{2} + 4 y^{2} + 2 z^{2}$ = $4 (x^{2} + 4 y^{2} + 2 z^{2})$

= $4 (x^{2} + 4 y^{2} + 2 z^{2})$ < $\frac{\in}{4}$

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