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The condition must hold for *every* $\epsilon > 0$ to ensure that $f(x)$ can be made arbitrarily close to $L$. If it only held for one specific $\epsilon$, the function values might be bounded within that range but not converge to $L$.

Conditions: $\epsilon$ represents an arbitrary positive tolerance.; The definition aims to prove convergence to a specific limit $L$.

The condition must hold for *every* $\epsilon > 0$ to ensure that $f(x)$ can be made arbitrarily close to $L$. If it only held for one specific $\epsilon$, the function values might be bounded within that range but not converge to $L$.

Conditions: $\epsilon$ represents an arbitrary positive tolerance.; The definition aims to prove convergence to a specific limit $L$.