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The formal definition of a limit, also known as the ε-δ definition, states the following: Given a function \(f(x)\), we say that \(\lim _{x \rightarrow a} f(x)=L\) if for any number \(\epsilon > 0\), there exists a number \(\delta > 0\) such that whenever \(0 < |x - a| < \delta\), it must be that \(|f(x) - L| < \epsilon\).

We can break down the definition into its main components: 1. \(\epsilon > 0\): ε (epsilon) represents any positive number, no matter how small. It is often interpreted as the desired "tolerance" or "accuracy" of the limit. 2. \(\delta > 0\): δ (delta) is another positive number that will be dependent on the given ε. It represents the "allowable" range of input values around the limit point 'a'. 3. \(0 < |x - a| < \delta\): This inequality states that the absolute difference between x and a must lie within the specified range set by δ, excluding the value of x=a (limit point) itself. 4. \(|f(x) - L| < \epsilon\): This inequality states that the absolute difference between the function's output and the limit value (L) must be smaller than the given ε. This ensures that the function approaches the desired limit L as x approaches a. The definition essentially says that as x approaches a, f(x) gets arbitrarily close to L, such that for any small positive tolerance ε, we can find some positive range of input values (δ) that produces output values that fall within that tolerance around L.