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.