Question

33. DISCUSS: A Function with Specified Limits Sketch the graph of an example of a function f that satisfies all of the following conditions.

$\begin{array}{l}\underset{x\to 0^{-}}{\lim }f\left(x\right)=2\underset{x\to 0^{+}}{\lim }f\left(x\right)=0\\ \underset{x\to 2}{\lim }f\left(x\right)=1f\left(0\right)=2f\left(2\right)=3\end{array}$

How many such functions are there

Short answer

One such graph is:

There are infinitely many such functions which follows these conditions.

Step-by-step solution

Step-1 – Given

Given conditions are:

$\begin{array}{l}\underset{x\to 0^{-}}{\lim }f\left(x\right)=2\underset{x\to 0^{+}}{\lim }f\left(x\right)=0\\ \underset{x\to 2}{\lim }f\left(x\right)=1f\left(0\right)=2f\left(2\right)=3\end{array}$

Step-2 – To determine

Sketch the graph of an example of a function f that satisfies all of the following conditions.

Step-3 – Calculation

Given conditions are:

$\begin{array}{l}\underset{x\to 0^{-}}{\lim }f\left(x\right)=2\underset{x\to 0^{+}}{\lim }f\left(x\right)=0\\ \underset{x\to 2}{\lim }f\left(x\right)=1f\left(0\right)=2f\left(2\right)=3\end{array}$

One such graph is:

There are infinitely many such functions which follows these conditions.