Give an example of continuously differentiable functions $f$,$g:\mathbb{R}^n\to\mathbb{R}$ for which $f(x_0)=g(x_0)=0$ and $\nabla f(x_0) = \lambda \nabla g(x_0) \neq 0$, for some point $x_0$ and number $\lambda$, but $$\lim\limits_{x\to x_0} \frac{f(x)}{g(x)}$$ does not exist.
Proof
Let the $i$th component of $g(x)$ be given by $g(x_i) = \frac{1}{x_{0_i}}(x_i^2-x_{0_i}^2)$.
Let the $i$th component of $f(x)$ be given by $f(x_i) = x_i-x_{0_i}$.
Then $f(x_0) = g(x_0) = 0$ and $\nabla g(x_0) = 2$, and $\nabla f(x_0) = 1$.
However, this limit is identical to $\lim\limits_{x\to 0} \frac{1}{x}$, which is known to not exist.
Something about this proof seems invalid to me.