Question

In Exercises \(55 - 58 ,\) complete parts \(( a ) - ( d )\) for the piecewise- definedfunction. \(\quad (\) a) Draw the graph of \(f\) . (b) At what points \(c\) in the domain of \(f\) does \(\lim _ { x \rightarrow c } f ( x )\) exist? (c) At what points \(c\) does only the left-hand limit exist? (d) At what points \(c\) does only the right-hand limit exist? $$f ( x ) = \left\\{ \begin{array} { l l } { \sin x , } & { - 2 \pi \leq x < 0 } \\ { \cos x , } & { 0 \leq x \leq 2 \pi } \end{array} \right.$$

Short answer

(a) The graph of \(f(x)\) can be sketched by plotting \(\sin x\) for \(-2\pi \leq x < 0\) and \(\cos x\) for \(0 \leq x \leq 2\pi\). (b) \(\lim _ { x \rightarrow c } f ( x )\) exists for every \(x\) in the domain of \(f\) i.e., \(-2\pi \leq x \leq 2\pi\) except at x=0. (c) Only the left-hand limit exists at x=0. (d) Only the right-hand limit exists at x=0.

Step-by-step solution

Drawing the graph

To draw the graph of the function, plot \(\sin x\) for interval \(-2\pi \leq x < 0\) and \(\cos x\) for interval \(0 \leq x \leq 2\pi\). The graph of \(\sin x\) for \(-2\pi \leq x < 0\) will start from the point \(( -2\pi, 0 )\) with one complete wave ending at the point \((0,0)\). The graph of \(\cos x\) for \(0 \leq x \leq 2\pi\) will start from \((0,1)\) and forms a full wave ending at the point \((2\pi,1)\).

Determining points with existing limits

The limit exists at points c in the domain of f where either both right hand and left-hand limits exist and are equal, or the function is defined at that point. Looking at the graphs, \(\lim _ { x \rightarrow c } f ( x )\) exists for every \(x\) in the domain of \(f\) i.e., \(-2\pi \leq x \leq 2\pi\) except at x=0.

Identifying points where only the left-hand limit exists

The left-hand limit of a function at a point exists if the function approaches a certain value as x approaches that point from the left. Looking at the graph, it is observed that only the left-hand limit exists at x=0.

Identifying points where only the right-hand limit exists

Similarly, the right-hand limit of a function at a point exists if the function approaches a certain value as x approaches that point from the right. Looking at the graph, it is observed that only the right-hand limit exists at x=0.