Problem 67 Graph each function and then fin... [FREE SOLUTION]

Chapter 1: Problem 67

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

Short Answer

Expert verified

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

Step by step solution

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.

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.

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.

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 $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檛 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.

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Short Answer

Step by step solution

Analyzing the Piecewise Function

Graphing the Function

Finding \( \lim_{x \to 0} H(x) \)

Finding \( \lim_{x \to 1} H(x) \)

Key Concepts

Piecewise Functions

Graphing Functions

Limit Existence

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