Question

43-45 Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f?

44. \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{{{\bf{2}}^x}}&{{\bf{if}}\,\,\,x \le {\bf{1}}}\\{{\bf{3}} - x}&{{\bf{if}}\,\,\,{\bf{1}} < x \le {\bf{4}}}\\{\sqrt x }&{{\bf{if}}\,\,\,x > {\bf{4}}}\end{array}} \right.\)

Short answer

Continuous at \(x = 4\) from the left.

Step-by-step solution

Sketch the graph of f

The figure below represents the graph of \(f\left( x \right)\).

Check the function \(f\left( x \right)\)

The function\(f\left( x \right)\)is continuous in the interval \(\left( { - \infty ,1} \right)\),\(\left( {1,4} \right)\), and \(\left( {4,\infty } \right)\).

Find the limit of the function at \(x = {\bf{1}}\)

Find the left-hand limit for \(f\left( x \right)\).

\(\begin{aligned}\mathop {\lim }\limits_{x \to {1^ - }} f\left( x \right) &= \mathop {\lim }\limits_{x \to {1^ - }} {2^x}\\ &= {2^1}\\ &= 2\end{aligned} \)

Find the right-hand limit for \(f\left( x \right)\).

\(\begin{{aligned}\mathop {\lim }\limits_{x \to {1^ + }} f\left( x \right) &= \mathop {\lim }\limits_{x \to {1^ + }} 3 - x\\ &= 3 - 1\\ &= 2\end{aligned}\)

As the left-hand limit and the right-hand limit are equal at \(x = 1\). So, the function is continuous at \(x = 1\).

Find the limit of the function at \(x = {\bf{4}}\)

Find theleft-hand limitfor \(f\left( x \right)\).

\(\begin{aligned}\mathop {\lim }\limits_{x \to {4^ - }} f\left( x \right) &= \mathop {\lim }\limits_{x \to {4^ - }} 3 - x\\ &= 3 - 4\\ &= - 1\end{aligned}\)

Find theright-hand limitfor \(f\left( x \right)\).

\(\begin{aligned} \mathop {\lim }\limits_{x \to {4^ + }} f\left( x \right) &= \mathop {\lim }\limits_{x \to {4^ + }} \sqrt x \\ &= \sqrt 4 \\ &= 2\end{aligned}\)

As the left-hand limit and the right-hand limit are not equal at \(x = 2\). So, the function is not continuous at in\(x = 2\).

But \(f\left( 4 \right) = - 1\), so f is continuous at \(x = 4\) from the left.