Frechet derivative limit

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 $f(cdot):mathbb{L}^2rightarrow mathbb{R}$, $xinmathbb{L}^2$ 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.