Let $\gamma_{\varepsilon} \rightharpoonup \gamma$ in $W^{1,\infty}(0,1)$. Then for any fixed $s \in \mathbb (0,1)$ does the limit $\lim_{\varepsilon \rightarrow 0} \frac{\gamma_{\varepsilon}(s\varepsilon)}{\varepsilon}$ exist?

I am on the fence as to whether or not it does. I rewrite it as $\lim_{\varepsilon \rightarrow 0} \frac{s\gamma_{\varepsilon}(s\varepsilon)}{s\varepsilon}$. Then possibly, the limit is $s D\gamma(0)$.