How do I graph this limit?

Question

How do I graph this limit ? The function has a limit as $x$ approaches $2$ of $5$ but $f(2) =4$. Specifically, what does $2$ of $5$ mean? The problem also wants me to write piecewise equations.

Answer 1

Your graph, whatever the function is will approach a $y$ value of 5 from both left and right sides but you will draw a point at the location $(2,4)$. The nomenclature you are using looks like this..

"The limit as x approaches 2 of $f(x)$ is 5" can also be written "the limit of $f(x)$ as x approaches 2 is 5" and is notationally written as $$\lim_{x\to2}f(x)=5$$

Answer 2

Your function is discontinuous at $x=2$. The graph would look like this in the sense that it will have a line, described by your function, which approaches a y-value of 5 in the neighborhood of $x=2$ but there will be a hole at $(2,5)$ and a lone point at $(2,4)$.

For a piecewise equation, you need to write the function as

$f(x) = \left\{ \begin{array}{lr} \text{your_function} & : x \neq 2\\ 4 & : x=2 \end{array} \right.$