l'Hopital's rule for 2 variables to compute Jacobian matrix

Question

I have a system of three ODEs and I have computed the Jacobian matrix.
One of the steady states is (0,0,0) and I am trying to linearize the system around this steady state. In the Jacobian matrix two of the terms are of the form $cx^2\over cx+y$ and $z\over y$.Here, $x,y,z$ are the variables of the system and $c$ is a known constant.

But when substituting x=0,y=0 and z=0 to the above terms it will be $0\over 0$.

So, in this case, can I use L'hospital rule for 2 variables to compute the Jacobian around that steady state?

That is I want to find
$\lim_{(x,y)\to (0,0)}{ cx^2\over cx+y}$ and $\lim_{(z,y)\to (0,0)} {z\over y}$.

I referred to the article on l'Hopital's rule for multi variable functions and with what it says in the article I could not find the limit of
$\lim_{(x,y)\to (0,0)}{ cx^2\over cx+y}$ and the limit of $\lim_{(z,y)\to (0,0)} {z\over y}$ does not exist.

Can someone please let me know a method to evaluate the two terms around the steady state.

Answer 1

With any limitaton in the domain the two limit doesn't exist, indeed simply note that for $t\to 0$

• $x=0 \quad y=t \implies {cx^2\over cx+y}=0$
• $x=t \quad y=-ct+t^2 \implies {cx^2\over cx+y}=\frac{ct^2}{ct-ct+t^2}=c$

and

• $z=0 \quad y=t \implies \frac z y = 0$
• $z=t \quad y=t \implies \frac z y = 1$