I was solving a problem by using the Frechet derivative to find the maximum. I result to the following limit
$$\lim_{||x||_{\mathbb{L}^2}\to 0}\frac{|f(x+t)-f(t)-J_{f(t)}(x)|_{\mathbb{R}}}{||x||_{\mathbb{L}^2}}=...=\underbrace{\lim_{||x||_{\mathbb{L}^2}\to 0}\frac{|\mathbb{E}(x^2)|_{\mathbb{R}}}{||x||_{\mathbb{L}^2}}}_{\text{why is this equal to $||x||_{\mathbb{L}^2}$}}=||x||_{\mathbb{L}^2}\to 0$$
where $f(\cdot):\mathbb{L}^2\rightarrow \mathbb{R}$, $x\in\mathbb{L}^2$ and $\mathbb{E}()$ is the expected value operator. Can anyone explain why this limit equals $||x||_{\mathbb{L}^2}$? I am a little confused! I do not provide the function $f$ and the linear functional $J$ for simplicity since there is no interest, but the solution of the problem results to this outcome.