Let $v_\delta$ be a sequence of continuously differentiable functions on $(-1,1)$ and $0\leq v_\delta\leq 1$. For each $\delta>0$, assume that $v_\delta(\delta)=v_\delta(-\delta)=1$ and $v_\delta(0)=0$. We also assume that $$ \limsup_{\delta\to 0}\left(\delta\int_{-\delta}^\delta|v_\delta'|^2dx\right)<\infty. $$ and $v_\delta'(0)=0$.
Would it be possible to prove that $$ \limsup_{\delta\to 0}\left(\delta\int_{-\xi_\delta}^{\xi_\delta}|v_\delta'|^2dx\right)=0? $$ Here $\xi_\delta$ denote the small $o$ notation, i.e., $$ \lim_{\delta\to 0}\frac{\xi_\delta}{\delta}=0. $$