Suppose we have a limit
$$\lim \limits_{x \to a} f(x) = f(a)$$
Using quantifer, the epsilon delta definition becomes
$$ \forall \varepsilon >0 \;\exists \delta >0 \; \forall x \in D \;\big( 0 < |x-a| < \delta \Rightarrow |f(x) - f(a) | < \varepsilon \big) $$
Now, I want to do change of variable in the limit definition. Let $ h = x-a $. So, the definition now becomes
$$ \lim \limits_{h \to 0} f(a+h) = f(a)$$
So, I want to know, how do I change the quntifier in the epsilon delta definition.
Thanks
More generally, $$\lim \limits_{\color{orange}x \to \color{red}a} \color{green}f(\color{orange}x) = \color{blue}L$$ is defined as $$\forall \varepsilon >0 \;\exists \delta >0 \; \forall \color{orange}x \in D_{\color{green}f} \;\big( 0 < |\color{orange}x-\color{red}a| < \delta \Rightarrow |\color{green}f(\color{orange}x) - \color{blue}L | < \varepsilon \big)$$ Accordingly, with $g(x):=f(a+x)$, $$\lim \limits_{\color{orange}h \to \color{red}0} \color{green}{f(a+\color{orange}x)} =\lim \limits_{\color{orange}h \to \color{red}0} \color{green}g(\color{orange}x) = \color{blue}{f(a)}$$ is $$\forall \varepsilon >0 \;\exists \delta >0 \; \forall \color{orange}h \in D_{\color{green}g} \;\big( 0 < |\color{orange}h-\color{red}0| < \delta \Rightarrow |\color{green}g(\color{orange}h) - \color{blue}{f(a)} | < \varepsilon \big)$$ or, $$\forall \varepsilon >0 \;\exists \delta >0 \; \forall \color{orange}h \in D_{\color{green}f}-a \;\big( 0 < |\color{orange}h| < \delta \Rightarrow |\color{green}{f(a+\color{orange}h)} - \color{blue}{f(a)} | < \varepsilon \big)$$