The problem is:

$\underset{(x,y,z)\to (0,0,0)}{lim}\frac{xy+2yz+3xz}{{x}^{2}+4{y}^{2}+9{z}^{2}}$

George Michael
2022-04-10
Answered

The problem is:

$\underset{(x,y,z)\to (0,0,0)}{lim}\frac{xy+2yz+3xz}{{x}^{2}+4{y}^{2}+9{z}^{2}}$

You can still ask an expert for help

ditumpasz9xj

Answered 2022-04-11
Author has **8** answers

Let's take the limit $z\to 0$ first, getting

$\underset{x,y\to 0}{lim}\frac{xy}{{x}^{2}+4{y}^{2}}$

Now consider what happens if you take the limit along$y=x$

$\underset{x,y\to 0}{lim}\frac{xy}{{x}^{2}+4{y}^{2}}=\underset{x\to 0}{lim}\frac{{x}^{2}}{5{x}^{2}}=\frac{1}{5}$

and along$y=2x$ :

$\underset{x,y\to 0}{lim}\frac{xy}{{x}^{2}+4{y}^{2}}=\underset{x\to 0}{lim}\frac{2{x}^{2}}{17{x}^{2}}=\frac{2}{17}$

That's enough to tell you that the limit depends on direction.

Now consider what happens if you take the limit along

and along

That's enough to tell you that the limit depends on direction.

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