Question

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

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

Short answer

For the value \(\mathop {\lim }\limits_{x \to 0} f\left( x \right) = 1, \mathop {\lim }\limits_{x \to {3^ - }} f\left( x \right) = - 2, \mathop {\lim }\limits_{x \to {3^ + }} f\left( x \right) = 2, f\left( 0 \right) = - 1, f\left( 3 \right) = 1\), the graph is presented below:

Step-by-step solution

One-sided limit

In one sided limit,

\(\mathop {\lim }\limits_{x \to {0^ - }} H(t) = 0{\rm{ }}and{\rm{ }}\mathop {\lim }\limits_{x \to {0^ + }} H(t) = 1\)

The symbol\(x \to {0^ - }{\rm{ and }}x \to {0^ + }{\rm{ }}\)indicates that considered values are less than zero and greater than zero respectively.

The negative sign (-) value will be gotten from the left side and the positive sign value (+) will be gotten from the right side.

If both the values will be equal then than the value of \(x \to 0\)will be also equal to both the values and if both the values are unequal then \(x \to 0\)won’t be equal to any of them.

Given information

The given data is,

\(\mathop {\lim }\limits_{x \to 0} f\left( x \right) = 1, \mathop {\lim }\limits_{x \to {3^ - }} f\left( x \right) = - 2, \mathop {\lim }\limits_{x \to {3^ + }} f\left( x \right) = 2, f\left( 0 \right) = - 1, f\left( 3 \right) = 1\)

Graph

The data will give the following graph for the value of \(x = - 1, - 2, 2, - 1, 1\) in the function, according to the one-sided limit explanation above.

Which is the required graph.