Problem 67 - Calculus and its applications

Question

Graph each function and then find the specified limits. When necessary, state that the limit does not exist. $$ \begin{array}{l} H(x)=\left\\{\begin{array}{ll} x+1, & \text { for } x<0 \\ 2, & \text { for } 0 \leq x<1 \\ 3-x, & \text { for } x \geq 1 \end{array}\right. \\ \text { Find } \lim _{x \rightarrow 0} H(x) \text { and } \lim _{x \rightarrow 1} H(x). \end{array} $$

Step-by-Step Solution

Answer

\( \lim_{x \to 0} H(x) \) does not exist; \( \lim_{x \to 1} H(x) = 2 \).

Step 1: Analyzing the Piecewise Function

The function is defined in three parts: \( H(x) = x+1 \) for \( x<0 \), \( H(x) = 2 \) for \( 0 \leq x < 1 \), and \( H(x) = 3-x \) for \( x \geq 1 \). Each segment defines a different expression for \( H(x) \) over different intervals. This means the behavior of \( H(x) \) depends heavily on which interval \( x \) belongs to.

Step 2: Graphing the Function

Create the graph based on each function segment's specific interval. For \( x < 0 \), graph the line \( y = x+1 \). For \( 0 \leq x < 1 \), the graph is a horizontal line at \( y = 2 \). For \( x \geq 1 \), plot the line \( y = 3-x \). The graph shows the continuity and transitions between these segments of the function.

Step 3: Finding \( \lim_{x \to 0} H(x) \)

For \( x \to 0^- \) (approaching from the left), \( H(x) = x+1 \), so \( \lim_{x \to 0^-} H(x) = 1 \). For \( x \to 0^+ \) (approaching from the right), \( H(x) = 2 \), so \( \lim_{x \to 0^+} H(x) = 2 \). Since the left-hand and right-hand limits are different, \( \lim_{x \to 0} H(x) \) does not exist.

Step 4: Finding \( \lim_{x \to 1} H(x) \)

For \( x \to 1^- \) (approaching from the left), \( H(x) = 2 \), so \( \lim_{x \to 1^-} H(x) = 2 \). For \( x \to 1^+ \) (approaching from the right), \( H(x) = 3-x \), and substituting \( x = 1 \) gives \( H(x) = 2 \), so \( \lim_{x \to 1^+} H(x) = 2 \). Both limits agree, so \( \lim_{x \to 1} H(x) = 2 \).

Key Concepts

Piecewise Functions

Piecewise functions are like stories with different chapters. They are a type of function which use different rules for different parts of their domain. Imagine a rulebook where certain rules apply only under specific conditions.
In mathematics, a piecewise function is defined by different expressions based on the input value (or interval) of the independent variable. These functions allow us to model scenarios where a single formula doesn’t fit all, such as systems with different operating modes or conditions.

• For example, in the piecewise function given, the rule changes at specific points: for values less than 0, it's defined as life gains (or losses) by adding 1; between 0 and 1, it's a constant value of 2; and, for values greater than or equal to 1, it gradually decreases with increasing values of \(x\).
• These functions are highly beneficial in real-world applications where a single mathematical expression cannot describe a complex scenario.

Recognizing the segments of a piecewise function is crucial for evaluating and understanding its behavior across its domain.

Graphing Functions

Graphing piecewise functions requires plotting each segment in its respective interval, which gives a visual representation of how the function behaves.

• Each interval of the piecewise function corresponds to a specific part of the graph. For instance, if your interval is \(x < 0\), you would graph the function using the equation associated with this interval, here \(y = x + 1\).
• The transition between these segments is indicated visually often with open or closed circles, showing whether the endpoint value is included (closed) or excluded (open) from the segment.
• When graphing, ensure each line or shape accurately reflects the mathematical description of that segment of the piecewise function. For example, in the piecewise function \(H(x)\), when \(0 \leq x < 1\), the graph shows a flat, horizontal line since the function remains constant at \(y = 2\).

Graphing is a powerful tool to quickly understand a function's dynamics and identify features like continuity and limits by observing how the segments fit together.

Limit Existence

Determining the existence of a limit as \(x\) approaches a certain point is crucial for understanding the behavior of piecewise functions at boundaries. When approaching these boundary points, we consider the function from the left and from the right.

• For a limit to exist at a certain point, \( \lim_{x \to c^-} f(x) \) and \( \lim_{x \to c^+} f(x) \) must be equal. If these two values are different, the limit at that point does not exist.
• In our example with the piecewise function \(H(x)\), when \(x\) approaches 0, the left-hand limit is different from the right-hand limit, hence \( \lim_{x \to 0} H(x) \) does not exist.
• On the other hand, as \(x\) approaches 1, both limits from the left and right are equal to 2, thus the limit does exist, and \( \lim_{x \to 1} H(x) = 2\).

Understanding limit existence not only assists in graph analysis but also in real-world applications where abrupt changes happen at specific points. It highlights discontinuities and can help us correct them if a smooth transition is required.