We know that the definition of a limit is: $\forall \epsilon > 0 \space \exists\delta>0, 0<|x-c|<\delta \to |f(x) - L| < \epsilon$
From my understanding, this means that the closer $x$ gets to $c$, then the closer $f(x)$ gets to $L$. However, since $x$ is never actually at $c$, then isn't it impossible for $f(x) = L$? If so, then why don't write $0<|x-c|<\delta \to 0<|f(x) - L| < \epsilon$ since it $f(x)$ can never equal $L$?