# The problem is: $$\displaystyle\lim_{{{\left({x},{y},{z}\right)}\rightarrow{\left({0},{0},{0}\right)}}}{\frac{{{x}{y}+{2}{y}{z}+{3}{x}{z}}}{{{x}^{{2}}+{4}{y}^{{2}}+{9}{z}^{{2}}}}}$$

The problem is:
$\underset{\left(x,y,z\right)\to \left(0,0,0\right)}{lim}\frac{xy+2yz+3xz}{{x}^{2}+4{y}^{2}+9{z}^{2}}$
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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.
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