I have been working on a projet regarding the wave equation, a part of which involves the derivation of the equation from a longitudinal wave in a string. At one point in the derivation, I have to show how the following limit reduces to the second partial derivative of y with respect to x:
$$\lim_{h\to 0}\frac{y(x+h;t)-2\cdot{y(x;t)}+y(x-h;t)}{h^{2}}=\frac{\partial^2}{\partial x^2}y(x;t)$$
My teacher had suggested to use L'Hôpital's rule twice. I found something similar in the wikipedia article about said rule, but in order to apply it my context I had to "redefine" the rule to express a partial derivative with respect to h of the numerator and denominator, i.e.:
$$\lim_{h\to 0}\frac{y(x+h;t)-2\cdot{y(x;t)}+y(x-h;t)}{h^{2}}\overset{L'H}=\lim_{h\to 0}\frac{\frac{\partial}{\partial h}(y(x+h;t)-2\cdot{y(x;t)}+y(x-h;t))}{\frac{\partial}{\partial h}(h^{2})}$$
In other words, I sort of extended L'Hôpital's rule to mean:
$$\lim_{x_{i}\to a}{_{\frac{0}{0}}{\frac{f(x_{1};...;x_{n})}{g(x_{1};...;x_{n})}}}\overset{L'H}=\lim_{x_{i}\to a}\frac{\frac{\partial}{\partial x_{i}}f(x_{1};...;x_{n})}{\frac{\partial}{\partial x_{i}}g(x_{1};...;x_{n})}$$
From here I was able to show the aforementioned equality, but I still feel a bit uneasy about the way I used L'Hôpital. I have not been able to find anything similar in my textbooks, nor online. As such, I want to know if using L'Hôpital in this manner is appropriate or if there is another way to obtain the same result.
Any help or insight is greatly appreciated, thank you.