> [!warning] Assumptions For interpreting the behaviour of a limit on a graph, assume the following: > - The limit as you approach from the right is $\displaystyle \lim_{ x \to n^+ } f(x)=a$ > - The limit as you approach from the left is $\displaystyle \lim_{ x \to n^+ } f(x)=b$ > - The function at $n$ is $f(n)=c$ > Where $a, b, c$ are defined [[Real Numbers|real values]] ($a,b,c\in \mathbb{R}$) ## Continuous Limits The continuous limit is a limit that is continuous by the [[Definition of Continuity|definition of continuity]] $ \begin{array}{c} \displaystyle \lim_{ x \to n^- }f(x) = a \\ \displaystyle \lim_{ x \to n^+ }f(x)=b \\ f(n)=a=b=c \end{array} $ ## Removable Discontinuity Limits The removable limit doesn't meet the requirements of the [[Definition of Continuity|definition of continuity]] because value of the function at the point is either undefined or not equal to the value of the limit. It a removable limit since a new function $g(x)$ can be defined at the point $x$ $ \begin{array}{c} \displaystyle \lim_{ x \to n^- }f(x) = a \\ \displaystyle \lim_{ x \to n^+ }f(x)=b \\ a=b \\ f(n)\neq a\neq b=c \quad (\text{c is either definied or is undefined ($\infty$)}) \end{array} $ ## Not Removable Discontinuity Limits If either the limit to a point or the function at the point is undefined. Then the limit is non-removable. $ \begin{array}{c} \text{$\displaystyle \lim_{ x \to n }f(x)= \infty$ or $f(n)=\infty$} \end{array} $ ## Jump Discontinuity Limits The jump discontinuity is when the limit from the left and right to a point are defined values, but they are not equal. The function can either be defined or undefined at this point. $ \begin{array}{c} \displaystyle \lim_{ x \to n^- } = a \\ \displaystyle \lim_{ x \to n^+ } = b \\ a \neq b \\ \\ f(n) =\infty \quad \text{or} \quad f(n)=c \end{array} $ > [!note] With jump discontinuity the function if defined can equal $a$ or $b$ but cannot equal both.