Question

Sketch the graph of an example of a function\(f\)that satisfies all of the given conditions.

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

Short answer

The graph of a function \(f\) that satisfies all of the given conditions is obtained.

Step-by-step solution

Intuitive Definition of One-Sided Limits

Write \(\mathop {\lim }\limits_{x \to {a^ - }} f\left( x \right) = L\) and it is called as theleft-hand limitof \(f\left( x \right)\) as \(x\) approaches \(a\) from the left is equal to \(L\). When we can make the values of \(f\left( x \right)\) arbitrarily close to \(L\) by restricting \(x\) to be sufficiently close to \(a\) with \(x\)less than\(a\).

Write\(\mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right) = L\)and it is called as aright-hand limit of \(f\left( x \right)\) as \(x\) approaches \(a\) from the right is equal to \(L\). When we can make the values of \(f\left( x \right)\) arbitrarily close to \(L\) by restricting \(x\) to be sufficiently close to \(a\) with \(x\)greater than\(a\).

Sketch the graph of an example of a function \(f\)

The explanation for the given conditions of a function\(f\)is shown below:

\(\mathop {\lim }\limits_{x \to - {3^ - }} f\left( x \right) = 3\). It follows that if\(x\)the approach\( - 3\)from the left-hand side is equal to 3, which leads to the point\(\left( { - 3,3} \right)\).

\(\mathop {\lim }\limits_{x \to - {3^ + }} f\left( x \right) = 2\). It follows that if\(x\)the approach\( - 3\)from the right is equal to 2, which leads to the point\(\left( { - 3,2} \right)\).

\(\mathop {\lim }\limits_{x \to {3^ - }} f\left( x \right) = - 1\). It follows that if\(x\)the approach\(3\)from the left is equal to\( - 1\), which leads to the point\(\left( {3, - 1} \right)\).

\(\mathop {\lim }\limits_{x \to {3^ + }} f\left( x \right) = 2\). It follows that if\(x\)the approach\(3\)from the right is equal to 2, which leads to the point\(\left( {3,2} \right)\).

\(\,f\left( { - 3} \right) = 2\,\,\,\,{\mathop{\rm and}\nolimits} \,\,\,f\left( 3 \right) = 0\). This leads to the points\(\left( { - 3,2} \right)\)and\(\left( {3,0} \right)\)is denoted in a thick dot.

Sketch the graph of an example of a function \(f\) that satisfies the above conditions as shown below:

Thus, the graph of the function \(f\)is obtained.