定義
![右極限](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![右極限](/img/7/277/wZwpmLxYTNyADO1gzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL4MzL1YzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
假設 是定義在區間 上的函式,如果下列準則成立:
![右極限](/img/a/325/wZwpmL2czMzMTO3QjN0kTO0UTMyITNykTO0EDMwAjMwUzL0YzLwQzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![右極限](/img/d/dd9/wZwpmL4UjN0EjMwITN2IDN0UTMyITNykTO0EDMwAjMwUzLyUzL4IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![右極限](/img/5/ead/wZwpmL1YzM1cTO5gzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4MzLwIzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![右極限](/img/1/036/wZwpmL3QzM5YjNzMjM0EDN0UTMyITNykTO0EDMwAjMwUzLzIzL3EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![右極限](/img/7/4b1/wZwpmL1EDO0YzNwAzM3QTN1UTM1QDN5MjM5ADMwAjMwUzLwMzL1AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
任意給定 ,能夠找到 ,使得滿足不等式 的一切 ,恆有 。
![右極限](/img/1/036/wZwpmL3QzM5YjNzMjM0EDN0UTMyITNykTO0EDMwAjMwUzLzIzL3EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![右極限](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![右極限](/img/f/513/wZwpmLyEDOygTNzAzMyADN0UTMyITNykTO0EDMwAjMwUzLwMzLzIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![右極限](/img/5/43f/wZwpmLxMDM2AzMycDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3AzL2EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
則稱當 由右邊趨於 時,收斂於極限 。記為 。
![右極限](/img/2/e31/wZwpmL3gDOxUDMyAzM3QTN1UTM1QDN5MjM5ADMwAjMwUzLwMzLwAzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![右極限](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![右極限](/img/f/513/wZwpmLyEDOygTNzAzMyADN0UTMyITNykTO0EDMwAjMwUzLwMzLzIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![右極限](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![右極限](/img/f/513/wZwpmLyEDOygTNzAzMyADN0UTMyITNykTO0EDMwAjMwUzLwMzLzIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![右極限](/img/5/43f/wZwpmLxMDM2AzMycDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3AzL2EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![右極限](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![右極限](/img/f/513/wZwpmLyEDOygTNzAzMyADN0UTMyITNykTO0EDMwAjMwUzLwMzLzIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![右極限](/img/1/036/wZwpmL3QzM5YjNzMjM0EDN0UTMyITNykTO0EDMwAjMwUzLzIzL3EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![右極限](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
數值 是 與 之間的距離,我們可以認為它是用 近似表示 所產生的誤差。因此 的定義,相當於斷言:用 近似表示 所產生的誤差可以小到我們任意指定的程度,只需要 從坐標充分靠近 。
![右極限](/img/b/9b7/wZwpmL0gTNzcDM0EDN3QTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL4QzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
性質
左極限與右極限統稱單側極限。
![右極限](/img/9/539/wZwpmL1YDMyMTN1MTNxUTN1UTM1QDN5MjM5ADMwAjMwUzLzUzLwQzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![右極限](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![右極限](/img/5/151/wZwpmL1gTM5cDO2YTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL2EzL1UzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![右極限](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![右極限](/img/5/151/wZwpmL1gTM5cDO2YTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL2EzL1UzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
①函式 當 時,極限存在,若且唯若函式在處左極限和右極限都存在,且兩者相等。用數學表達式表示為:
![右極限](/img/1/a83/wZwpmLzITN3UTNzMTOwMzM1UTM1QDN5MjM5ADMwAjMwUzLzkzLwczLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![右極限](/img/a/9f1/wZwpmLxQTO1UjMxQTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL1UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![右極限](/img/a/b31/wZwpmL3EDO4MTN0YDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2AzL3AzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![右極限](/img/6/6d8/wZwpmLzEDNwkTO0YjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2IzL2IzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![右極限](/img/5/945/wZwpmL2gjMyQDM3cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzL2MzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
存在和都存在且。
②函式的左極限和右極限不一定相等,例如:
![右極限](/img/3/69b/wZwpmLzEzN3QDM4gzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4MzLygzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![右極限](/img/e/5df/wZwpmLxEDMyMDN0cDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3AzLxMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
此時稱函式在該點有“跳躍”。
③左極限與右極限只要有其中有一個極限不存在,則函式在該點極限不存在。