> [!summary] The epsilon delta definition of a limit is a more accurate definition of what it means to approach a value > **Key Result:** > $ \begin{array}{c} \text{For any distance $\varepsilon >0$ there is a distance $\delta>0$ such that if:} \\ \\ 0<|x-a|<\delta \quad (\text{There is a range of input values that follow this rule}) \\ \\ \text{Then:}\\ |f(x)-L| < \varepsilon \end{array} > $ >[!info]+ Read Time **⏱ 2 mins** # Definition The epsilon-delta definition of a [[Limits|limit]] is a more rigorous definition for approaching a value. It explains the mathematical reasoning for why the $\displaystyle\lim_{ x \to a } f(x)=L$ or when the approaching value converges to $L$ ## Explanation > [!warning] Assumptions The mathematical definition uses an epsilon-delta approach. To understand this, assume the following: > - The definition of a limit that converges is $\displaystyle\lim_{ x \to a } f(x)=L$ The definition of a convergent limit is that the limit close to $a$ is very similar to $L$ not matter the direction you approach it from. Mathematically it denoted by a vertical distance away from $L$ called $\varepsilon$. But this is taking into account the output of a function. Remember that the limit approaches to $a$ so mathematically the horizontal distance away from c is called $\delta$. This takes into account out input to the function. ![[edd_1.png|500]] So the following requirements is made to prove that a function is convergent. $ \begin{array}{c} \text{For any distance $\varepsilon >0$ there is a distance $\delta>0$ such that if:} \\ \\ 0<|x-a|<\delta \quad (\text{There is a range of input values that follow this rule}) \\ \\ \text{Then:}\\ |f(x)-L| < \varepsilon \end{array} $ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/JbbRaiXI6yw?si=MVPaGzguBHnxRQ5z" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe> <iframe width="560" height="315" src="https://www.youtube.com/embed/8qi-hv48Hws?si=JcSafsvaWwDNm4gK" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe> <iframe width="560" height="315" src="https://www.youtube.com/embed/kfF40MiS7zA?si=WxEMscB9JkFW4Lan" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>