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? $$f ( x ) = \left\\{ \begin{array} { l l } { x , } & { - 1 \leq x < 0 , \text { or } 0 < x \leq 1 } \\ { 1 , } & { x = 0 } \\ { 0 , } & { x < \- 1 , \text { or } x > 1 } \end{array} \right.$$

Short answer

The piecewise function is graphed with parts defined on intervals \( x < -1, -1 \leq x < 0, 0, 0 < x \leq 1, x > 1 \). The limit exists for all x except at x = 0. Only the left-hand limit exists at x = 0; there are no points with only a right-hand limit.

Step-by-step solution

- Graphing the Function

Since this is a piecewise function, it is defined in sections. Graph the function as per these defined intervals. For \( -1 \leq x < 0 \) and \( 0 < x \leq 1 \), plot the line \( f(x) = x \). At \( x = 0 \), plot the point \( (0,1) \). For \( x < -1 \) and \( x > 1 \), plot the lines \( f(x) = 0 \). This will give a visual representation of the function.

- Identifying Points where Limits Exist

This step involves identifying points at which \( \lim _ { x \rightarrow c } f(x) \) exists. A limit exists at a particular point when the left-hand limit and right-hand limit both converge to the same value. Checking all points within the function's domain, we see the limit exists for all x except at x = 0.

- Identifying Points with Only Left-Hand Limits

To identify points where only the left-hand limit exists, we need to consider points where the function is defined from the left but not the right. For this function, the left-hand limit only exists at x = 0.

- Identifying Points with Only Right-Hand Limits

Similarly, to identify points where only the right-hand limit exist, we need to look for points where the function is defined from the right. In this case, there are no points which have only right-hand limit since at x = 0 it is defined from both left and right.

Key concepts

Graphing Piecewise Functions

When it comes to graphing piecewise functions, visualize it as a puzzle; each piece is like a segment with its own rule.

The first step is to create a separate graph for each segment based on its interval and function rule. This includes plotting points and drawing lines or curves for the function within the specified intervals. For example, in the given function, for the interval \( -1 \leq x < 0 \) and \( 0 < x \leq 1 \) the plot would be a straight line, given by \( f(x) = x \). At \( x = 0 \), you plot a single point at \( (0,1) \). The segments \( x < -1 \) and \( x > 1 \) would also be straight lines, but they would be constant at \( f(x) = 0 \).

Remember to use open circles to show that a point is not included in the interval and filled circles when it is included. Proper notation and accuracy in your graph will ensure a clear visual representation of the piecewise function.

Limits of Piecewise Functions

Understanding limits of piecewise functions is crucial in calculus, as they depict the behavior of the function as it approaches a certain point.

To determine limits in a piecewise function, look at the behavior of the function as it gets close to the 'cutoff' points between pieces. If both sides of the pieces approach the same value as \( x \) approaches the cutoff point, the limit exists at that point. For the given function, the limit exists at all points except \( x = 0 \), as there is a 'jump' in the function due to the differing value at that point.

Make sure you examine the function at each interval: do the left-hand side and right-hand side approach the same value as they get closer to the dividing point? If yes, the limit at that point exists.

Left-Hand Limit

The left-hand limit describes the value your function approaches as you get close to a point \( c \) from the left side.

For the given function, to find the left-hand limit at \( x = 0 \), you would look at the segment for \( -1 \leq x < 0 \) and observe the behavior of \( f(x) \) as \( x \) approaches 0 from the left side. You should find that the left-hand limit at \( x = 0 \) is 0.

Keep in mind, left-hand limits are useful when assessing continuity and for understanding the behavior of the function at a boundary of its domain, especially when the function has a different definition on either side of the boundary.

Right-Hand Limit

Conversely, the right-hand limit is about the approach from the right. It determines what value the function heads toward as \( x \) approaches a point \( c \) from the right side.

In the context of the given exercise, look at the piece given by \( 0 < x \leq 1 \) and see what happens as \( x \) moves towards 0 from the right. You would notice that the function \( f(x) \) approaches the value of 0. However, for the point \( x = 0 \) itself, since it's indicated that \( f(0) = 1 \) for our specific function, there's a discontinuity at that point.

It's key to note that for continuous points, the left-hand and right-hand limits should match, affirming that the full limit exists at the point. The given function lacks right-hand limits only in situations where approaching from the right is not defined or leads to a different value compared to left-hand approach, which does not occur in this particular function.