The $\varepsilon$-$\delta$ definition of a limit is a mathematical definition that is used to precisely define the concept of a limit in calculus. It states that for a function $f(x)$, the limit of $f(x)$ as $x$ approaches a specific value $c$ is equal to $L$ if for any positive value of $\varepsilon$, there exists a positive value of $\delta$ such that when the distance between $x$ and $c$ is less than $\delta$, the distance between $f(x)$ and $L$ is less than $\varepsilon$.
The $\varepsilon$-$\delta$ definition is important because it provides a rigorous and precise definition of the concept of a limit in calculus. It allows us to prove the existence of limits and their values, and it is the foundation for many important theorems and techniques in calculus.
To use the $\varepsilon$-$\delta$ definition to prove a limit, you must first determine the limit you are trying to prove. Then, you must choose a value for $\varepsilon$ and use algebraic manipulations to find a corresponding value for $\delta$ that satisfies the definition. Finally, you must show that for any value of $\varepsilon$, there exists a value of $\delta$ that satisfies the definition.
Yes, the $\varepsilon$-$\delta$ definition can be used to prove the continuity of a function. If a limit exists for a function $f(x)$ at a specific point $c$, and the value of the limit is equal to $f(c)$, then $f(x)$ is continuous at $c$ according to the $\varepsilon$-$\delta$ definition.
One limitation of the $\varepsilon$-$\delta$ definition is that it only applies to functions that are defined at a specific point $c$. It cannot be used to prove the existence or non-existence of limits at points where a function is not defined. Additionally, it can be difficult to use the definition in more complex situations, such as when dealing with multi-variable functions or sequences.