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The \( \varepsilon \)-\( \delta \) definition of a limit for \( \lim_{x \to c} f(x) = L \) is as follows:

For every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - c| < \delta \), then \( |f(x) - L| < \varepsilon \).

In this formal definition:

- \( \varepsilon \) (epsilon) represents how close \( f(x) \) is to \( L \),
- \( \delta \) (delta) represents how close \( x \) is to \( c \),
- The expression says that as \( x \) gets closer to \( c \), the function \( f(x) \) gets arbitrarily close to \( L \).

Would you like more details or have any further questions?

Here are 5 related questions:
1. How can the \( \varepsilon \)-\( \delta \) definition be applied to a specific function?
2. What is the purpose of introducing \( \varepsilon \) and \( \delta \) in the limit definition?
3. Can you explain the difference between one-sided and two-sided limits using \( \varepsilon \)-\( \delta \)?
4. How does the \( \varepsilon \)-\( \delta \) definition help in proving limits rigorously?
5. How is the definition adapted for limits approaching infinity?

**Tip**: Visualizing the \( \varepsilon \)-\( \delta \) condition on a graph can help in understanding how the function behaves near the limit.

Please write the ε-δ definition for lim (x→c) f(x) = L.

Understand the formal ε-δ definition of limits in calculus, including the rigorous condition for lim (x→c) f(x) = L. This covers key concepts in limits, continuity, and calculus proofs.

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