What is the value of $\lim_{(x,y,z)\rightarrow(0,0,0)}\log(\frac{x}{yz})$?

Question

What is the value of $\lim_{(x,y,z)\rightarrow(0,0,0)}\log(\frac{x}{yz})$?

(a)$0$

(b)$2$

(c)$4$

(d)does not exist

Let us take $y=mx$ and $z=mx$ where $m$ is an arbitrary parameter,then

$\lim_{(x,y,z)\rightarrow(0,0,0)}\log(\frac{x}{yz})=\lim_{m\rightarrow 0}\log(\frac{1}{m^2})=$does not exist

Is it correct??

Answer 1

Note that the expression is well defined for $x/yz >0$, in this case

• for $x=y=z=t$ with $t\to 0^+$

$$\log(\frac{x}{yz})=\log(\frac{1}{t})\to -\infty$$

• for $x=t^2,\,y=z=t$ with $t\to 0^+$

$$\log(\frac{x}{yz})=\log(1)\to 0$$

thus the answer is correct since

$$\lim_{(x,y,z)\rightarrow(0,0,0)}\log(\frac{x}{yz})$$

doesn't exist.