I'm actually having questions over applying Taylor expansion to apply the L'Hospital rule.
Let $f,g : \mathbb{R}^n \to \mathbb{R}$ function
Since we know that for a $C^1$ function such that $f(y) = g(y) =0$, and $df(x) =\lambda dg(x)$. We can show that $\lim_{x\to y}\frac{f(x)}{g(x)} = \frac{0 + df(y)(x-y)+o(|x-y|) }{0 + dg(y)(x-y)+o(|x-y|)}= \frac{\lambda dg(x)(x-y|)}{dg(x)(x-y} = \lambda$.
But obviously, such a scenario won't work for many cases since the limit might not even exist, say $f = x-y$ and $g = sin(x) - sin(y)$.
Why would such a thing would happen?