Question

In Exercises \(51 - 54 ,\) complete parts \(( a ) , (\) b) \(,\) and \(( c )\) for the piecewise-defined function. (a) Draw the graph of \(f .\) (b) Determine \(\lim _ { x \rightarrow c ^ { + } } f ( x )\) and \(\lim _ { x \rightarrow c ^ { - } } f ( x )\) (c) Writing to Learn Does \(\lim _ { x \rightarrow c } f ( x )\) exist? If so, what is it? If not, explain. $$c = 2 , f ( x ) = \left\\{ \begin{array} { l l } { 3 - x , } & { x < 2 } \\\ { \frac { x } { 2 } + 1 , } & { x > 2 } \end{array} \right.$$

Short answer

The graph of the piecewise function will be two separate lines intersecting at x = 2. The left limit of f(x) as x approaches from the left side (x < 2) in \(3-x\) is 1, while the right limit of f(x) as x approaches from the right side (x > 2) in \(\frac{x}{2} + 1\) is 2. Since both limits do not equal, the limit as \(x \rightarrow 2\) does not exist.

Step-by-step solution

Draw the Graph

Draw two separate graphs for each function, \(3 - x\) and \(\frac{x}{2} + 1\), respectively. But remember, \(3 - x\) is only valid for x < 2 and \(\frac{x}{2} + 1\) is valid for x > 2. Evaluate the functions at x=2 to ensure continuity.

Determine the Right and Left Limits

Determine the left limit as \(x \rightarrow c \; (x \rightarrow 2^{-})\) and right limit as \(x \rightarrow c \; (x \rightarrow 2^{+})\). For the left limit, substitute \(x = 2\) in \(3-x\) to get 1 and for the right limit, substitute \(x = 2\) in \(\frac{x}{2} + 1\) to get 2.

Determine the Existence of Limit

The overall limit as \(x \rightarrow c (x \rightarrow 2)\) exists only if the left limit equals the right limit. In this case, the limits are not equal (1 ≠ 2), therefore the limit as \(x \rightarrow 2\) does not exist.