Question

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

16.\(\mathop {\lim }\limits_{x \to 0} f\left( x \right) = 4,\,\,\mathop {\lim }\limits_{x \to {8^ - }} f\left( x \right) = 1,\,\,\,\mathop {\lim }\limits_{x \to {8^ + }} f\left( x \right) = - 3,\,\,f\left( 0 \right) = 6,\,\,\,\,f\left( 8 \right) = - 1\).

Short answer

The graph of a function \(f\) that satisfies all of the given conditionsis 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 iscalled 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 condition is shown below:

\(\mathop {\lim }\limits_{x \to 0} f\left( x \right) = 4\). It follows that if\(x\)approach 0 from either the right or left-hand side is equal to 4, which leads to the point\(\left( {0,4} \right)\).

\(\mathop {\lim }\limits_{x \to {8^ - }} f\left( x \right) = 1\). It follows that if\(x\)approach 8 from the left is equal to 1, which leads to the point\(\left( {8,1} \right)\).

\(\mathop {\lim }\limits_{x \to {8^ + }} f\left( x \right) = - 3\). It follows that if\(x\)the approach\(8\)from the right is equal to\( - 3\), which leads to the point\(\left( {8, - 3} \right)\).

\(f\left( 0 \right) = 6\)and\(f\left( 8 \right) = - 1\). This leads to the points\(\left( {0,6} \right)\)and\(\left( {8, - 1} \right)\)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.