Question

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

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

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 the left-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 a right-hand limitof \(f\left( x \right)\) 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 explanations for the given conditions of a function \(f\) as shown below:

\(\mathop {\lim }\limits_{x \to - {1^ - }} f\left( x \right) = 0\). It follows that if \(x\)the approach \( - 1\) from the left side is equal to 0, which leads to the point \(\left( { - 1,0} \right)\).

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

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

Also, \(f\left( { - 1} \right) = 2\,\,{\mathop{\rm and}\nolimits} \,\,f\left( 2 \right) = 1\). This leads to the points \(\left( { - 1,2} \right)\) and \(\left( {2,1} \right)\)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.