Problem 34 - Precalculus: Mathematical for Calculus

Question

Let \(h(x)=\left\\{\begin{array}{ll}x & \text { if } x < 0 \\ x^{2} & \text { if } 0 < x \leq 2 \\ 8-x & \text { if } x > 2\end{array}\right.\) (a) Evaluate each limit, if it exists. (i) \(\lim _{x \rightarrow 0^{+}} h(x)\) (iv) \(\lim _{x \rightarrow 2^{-}} h(x)\) (ii) \(\lim _{x \rightarrow 0} h(x)\) (v) \(\lim _{x \rightarrow 2^{+}} h(x)\) (iii) \(\lim _{x \rightarrow 1} h(x)\) (vi) \(\lim _{x \rightarrow 2} h(x)\) (b) Sketch the graph of \(h\)

Step-by-Step Solution

Answer

Limits: (i) 0, (ii) 0, (iii) 1, (iv) 4, (v) 6, (vi) does not exist. Graph: segments from the piecewise function.

Step 1: Understanding the Piecewise Function

The function \( h(x) \) is defined piecewise with three different expressions based on the values of \( x \). We need to evaluate the limits approaching 0, 1, and 2, considering the appropriate expressions for \( h(x) \) in each case.

Step 1: Evaluate \( \lim_{x \to 0^+} h(x) \)

For \( x > 0 \) and approaching 0 from the positive side, \( h(x) = x^2 \). Therefore, \( \lim_{x \to 0^+} h(x) = \lim_{x \to 0^+} x^2 = 0^2 = 0 \).

Step 2: Evaluate \( \lim_{x \to 0} h(x) \)

Since \( h(x) = x^2 \) for \( 0 < x \leq 2 \) and \( h(x) = x \) for \( x < 0 \), both approach 0 as \( x \to 0 \). Hence, \( \lim_{x \to 0} h(x) = 0 \).

Step 3: Evaluate \( \lim_{x \to 1} h(x) \)

For \( x \) approaching 1, we use \( h(x) = x^2 \), since \( 0 < x \leq 2 \). Therefore, \( \lim_{x \to 1} h(x) = \lim_{x \to 1} x^2 = 1^2 = 1 \).

Step 4: Evaluate \( \lim_{x \to 2^-} h(x) \)

For \( x \) approaching 2 from the left, \( h(x) = x^2 \). Therefore, \( \lim_{x \to 2^-} h(x) = \lim_{x \to 2^-} x^2 = 2^2 = 4 \).

Step 5: Evaluate \( \lim_{x \to 2^+} h(x) \)

For \( x \) approaching 2 from the right, \( h(x) = 8 - x \). Hence, \( \lim_{x \to 2^+} h(x) = \lim_{x \to 2^+} (8 - x) = 8 - 2 = 6 \).

Step 6: Evaluate \( \lim_{x \to 2} h(x) \)

Since \( \lim_{x \to 2^-} h(x) = 4 \) and \( \lim_{x \to 2^+} h(x) = 6 \), the two one-sided limits are not equal, so \( \lim_{x \to 2} h(x) \) does not exist.

Step 7: Sketching the Graph of \( h \)

The graph features three segments: a line through the origin for \( x < 0 \), a parabola \( y = x^2 \) from \( 0 < x \leq 2 \), and a downward-sloping line \( y = 8 - x \) for \( x > 2 \). Note the open boundaries at \( x = 0 \) and \( x = 2 \).

Key Concepts

Understanding Piecewise Functions

A piecewise function is a function that has different expressions for different intervals of the domain. In our exercise, the function \( h(x) \) is defined in three parts, each covering a separate range of \( x \) values. This is like having three small, different functions stitched together, each operating over a distinct part of \( x \) values.

• For \( x < 0 \), \( h(x) = x \)
• For \( 0 < x \leq 2 \), \( h(x) = x^2 \)
• For \( x > 2 \), \( h(x) = 8 - x \)

When you evaluate such functions, it's important to choose the correct expression based on where \( x \) is in the domain. This influences the resultant value and is crucial for calculating limits.

Exploring One-sided Limits

One-sided limits refer to the limits of a function as \( x \) approaches a specific point from one side—either from the right or the left. In piecewise functions, these often show us how the function behaves near the boundaries where its expressions change.

• The right-hand limit \( \lim_{x \to a^+} h(x) \) checks what \( h(x) \) approaches as \( x \) tends to \( a \) from values greater than \( a \).
• The left-hand limit \( \lim_{x \to a^-} h(x) \) examines what \( h(x) \) approaches as \( x \) tends to \( a \) from values less than \( a \).

To find if a general limit \( \lim_{x \to a} h(x) \) exists, the one-sided limits must both exist and equal each other. For example, if they differ, as seen at \( x = 2 \) in our exercise, the two sides don't meet, and thus the limit doesn't exist. This showcases an important aspect of continuity and limits.

Graph Sketching Techniques

Sketching the graph of piecewise functions can be a creative process, mapping out separate parts to see the overall picture. Here are a few steps to help guide you:

• Start with each separate function segment. For \( x < 0 \), draw the line \( y = x \), which passes through the origin. This is straightforward since it's a simple line.
• For \( 0 < x \leq 2 \), sketch the parabola \( y = x^2 \). It smoothly curves upwards from the origin and through to \( x = 2 \).
• For \( x > 2 \), draw the line \( y = 8 - x \). This line slopes downwards, intersecting the y-axis at \( y = 8 \).

To mark transitions between segments, note open and closed circles where appropriate. An open circle indicates a point not included in that segment if the interval does not include it, while a closed circle is used when the end point is part of the interval. These graphical parts form the complete view of \( h(x) \). Understanding each segment visually helps in comprehending the piecewise nature of the function.