Let $\lambda \in \mathbb{R}$ and Let $A$ be a linear operator on $\mathbb{R}^3$. Define $f:\mathbb{R}^3 \to \mathbb{R}$ by $f(x):=\langle A(x),x\rangle- \lambda \langle x,x\rangle $ where $\langle,\rangle$ is the usual dot product on $\mathbb{R}^3$.
Does $\lim_{x\rightarrow0}\frac{f(x)}{|x|}$ exist?
My computation: $\begin{align*} \lim_{x\rightarrow0}\frac{f(x)}{|x|} &=\lim_{x\rightarrow0}\frac{\langle A(x)-\lambda x,x\rangle}{|x|} \\ &=\lim_{x\rightarrow0}\frac{|A(x)-\lambda x||x|\cos \theta_{x}}{|x|} \\ &=\lim_{x\rightarrow0}|A(x)-\lambda x|\cos \theta_{x}=0\end{align*}$ where $\theta_{x}$ is the angle between $A(x)-\lambda x$ and $x$. I wonder whether $\theta_{x}$ is continuous? If yes, then my computation is ok.