Problem 68 - 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} G(x)=\left\\{\begin{array}{ll} 2+x, & \text { for } x \leq-1 \\ x^{2}, & \text { for }-1

Step-by-Step Solution

Answer

\( \lim_{x \to -1} G(x) = 1 \), \( \lim_{x \to 3} G(x) = 9 \).

Step 1: Understand the Piecewise Function

The function \( G(x) \) is defined as three different expressions based on the value of \( x \):- \( G(x) = 2 + x \) when \( x \leq -1 \)- \( G(x) = x^2 \) for \( -1 < x < 3 \)- \( G(x) = 9 \) for \( x \geq 3 \)

Step 2: Evaluate \(\lim_{x \to -1} G(x)\)

For \( x = -1 \), we use the expression from the condition \( x \leq -1 \), which gives \( G(x) = 2 + x = 2 - 1 = 1 \).For \( x \) approaching -1 from the right (\( x > -1 \)), \( G(x) = x^2 \). Evaluating this as \( x \) approaches -1, we get \( (-1)^2 = 1 \).Since both the left-hand and right-hand limits give the same result,\[ \lim_{x \to -1} G(x) = 1 \].

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

For \( x \) approaching 3 from the left (\( x < 3 \)), \( G(x) = x^2 \). Evaluating this as \( x \) approaches 3, we have \( 3^2 = 9 \).For \( x = 3 \) or \( x > 3 \), \( G(x) = 9 \).Since both the left-hand and right-hand limits give the same result for approaching \( x = 3 \),\[ \lim_{x \to 3} G(x) = 9 \].

Step 4: Check for Consistency and Sketch the Graph

To ensure the correctness, note that the limits as \( x \rightarrow -1 \) and \( x \rightarrow 3 \) match the function values at those points, where applicable. The function should be sketched with a straight line for \( x \leq -1 \), a parabola for \( -1 < x < 3 \), and a constant line at 9 for \( x \geq 3 \).

Key Concepts

Limits

In mathematics, **limits** help us understand the behavior of functions as they approach a particular point. Specifically, limits allow us to examine what value a function approaches as the independent variable (here, **x**) gets closer to a certain number.
For piecewise functions like \( G(x) \), limits are crucial in determining how the function behaves differently at distinct segments of \( x \) values.

• The **left-hand limit** concerns the behavior of \( G(x) \) as **x** approaches a point from the left (i.e., values less than the point).
• The **right-hand limit** involves how \( G(x) \) behaves as **x** approaches from the right (i.e., values greater than the point).

When both of these limits are equal, the general limit exists at that point. In our example:

• For \( x \to -1 \), both left-hand and right-hand limits equal 1, so \( \lim_{x \to -1} G(x) = 1 \).
• For \( x \to 3 \), both limits equal 9, so \( \lim_{x \to 3} G(x) = 9 \).

Understanding limits is essential in evaluating and interpreting piecewise functions effectively.

Graphing Functions

Graphing functions, especially piecewise ones, provides a visual representation of how these functions behave over different ranges of values. For the function \( G(x) \), graphing involves using multiple expressions that define \( G(x) \) across different segments:

• For \( x \leq -1 \): Graphing \( 2+x \) results in a straight line, showcasing a point-slope form representation.
• For \( -1 < x < 3 \): This section is represented by \( x^2 \), a quadratic curve known as a parabola.
• For \( x \geq 3 \): The graph is a horizontal line fixed at \( y = 9 \).

To effectively piece these segments together for \( G(x) \), we avoid connecting different parts, unless they meet at points of intersection.
So, ensuring the appropriate limits and boundaries are visualized correctly helps reinforce understanding and interpretation of function behaviors. Well-constructed graphs serve as a crucial tool in understanding not only general behavior but also specific details, such as intercepts and changes in slope.

Continuity in Calculus

**Continuity** in calculus determines if a function is smooth and unbroken at a particular point or across an interval. For a function to be continuous at a point \( x=c \), three conditions must be met:

• First, \( G(c) \) must be defined.
• Second, \( \lim_{x \to c} G(x) \) must exist.
• Lastly, \( \lim_{x \to c} G(x) = G(c) \).

Applying this to our piecewise function \( G(x) \):

• At \( x = -1 \), the limits from both sides equal the function value, ensuring continuity.
• At \( x = 3 \), \( \lim_{x \to 3} G(x) = 9 \) matches the function's value, confirming continuity here as well.

Overall, piecewise functions often demonstrate continuity within segments but can have breaches at their breakpoints depending on the conditions specified. Understanding how continuity works with limits and evaluating specific points helps solidify a foundational calculus concept.