Problem 12 - Calculus: Early Transcendentals

Question

Sketch the graph of the function and use it to determine the values of \( a \) for which \( \displaystyle \lim_{x\to a}f(x) \) exists. \( f(x) = \left\\{ \begin{array}{ll} 1 + \sin x & \mbox{if \) x < 0 \(}\\\ \cos x & \mbox{if \) 0 \le x \le \pi \(}\\\ \sin x & \mbox{if \) x > \pi \(} \end{array} \right.\)

Step-by-Step Solution

Answer

The limit exists for all \( a \neq \pi \).

Step 1: Analyze the Function and Intervals

Begin by examining the piecewise function \( f(x) \). It's defined differently over three specific intervals: 1. \( f(x) = 1 + \sin x \) for \( x < 0 \) 2. \( f(x) = \cos x \) for \( 0 \le x \le \pi \) 3. \( f(x) = \sin x \) for \( x > \pi \). Each segment corresponds to a specific trigonometric function, adjusted or translated accordingly.

Step 2: Sketch Each Segment of the Function

Next, graph each segment over its specified domain. - For \( x < 0 \), \( f(x) = 1 + \sin x \) is a vertical translation of \( \sin x \). Sketch this for values less than zero.- For \( 0 \le x \le \pi \), sketch \( \cos x \), a straightforward cosine wave.- For \( x > \pi \), sketch \( \sin x \), focusing on values greater than \( \pi \).Ensure that each segment connects smoothly within its defined interval.

Step 3: Identify Potential Points of Discontinuity

Identify points where there's a change in definition which could lead to discontinuity, namely: \( x = 0 \) and \( x = \pi \). Also, consider infinity boundaries for completeness. These are the critical points where you need to check whether a limit exists.

Step 4: Evaluate Limits at Critical Points

Evaluate the limit as \( x \to 0 \) and \( x \to \pi \):- As \( x \to 0^- \), the function follows \( 1 + \sin x \) which approaches 1.- As \( x \to 0^+ \), the function follows \( \cos x \) which is 1.- As \( x \to \pi^- \), the function remains \( \cos x \) which becomes -1.- As \( x \to \pi^+ \), the function follows \( \sin x \) and approaches 0.

Step 5: Determine Continuity and Existence of Limits

From Step 4, confirm where the left-hand and right-hand limits are equal:- At \( x = 0 \), the limit exists because \( \lim_{x\to 0^-} f(x) = \lim_{x\to 0^+} f(x) = 1 \).- At \( x = \pi \), the limit does not exist as \( \lim_{x\to \pi^-} f(x) = -1 \) and \( \lim_{x\to \pi^+} f(x) = 0 \) are not equal.The limit exists for \( a eq \pi \).

Step 6: Conclusion on Range of 'a' for Limit Existence

The limit \( \lim_{x \to a} f(x) \) exists for any \( a eq \pi \) within the function's domain, as there are no discontinuities due to smoothly joined pieces apart from at \( x = \pi \).

Key Concepts

Piecewise Functions

A piecewise function is a function that is defined by different expressions, each of which applies to a different part of the domain. In the given problem, the piecewise function is divided into three parts, each defined over specific intervals of the input variable, \( x \). These intervals are:

• \( f(x) = 1 + \sin x \) for \( x < 0 \)
• \( f(x) = \cos x \) for \( 0 \le x \le \pi \)
• \( f(x) = \sin x \) for \( x > \pi \)

Understanding piecewise functions requires examining how each defined part behaves over its domain. Each component is influenced by trigonometric functions (sine and cosine), which are influential in determining the behavior and continuity of the overall function. It’s crucial to ensure that at the endpoints and boundaries where the defining expressions switch, the function remains continuous unless a discontinuity is inherently a part of the function’s nature. To evaluate limits and sketch graphs, identifying and making sense of these transition points between different parts is essential.

Graph Sketching

Graph sketching involves plotting each segment of a piecewise function over its domain to visualize how the function behaves. For the provided exercise, each section of the function represents a familiar trigonometric shape:

• For \( x < 0 \), \( f(x) = 1 + \sin x \) is a sine wave shifted upward by 1 unit. This translation reflects changes in amplitude and vertical position.
• For \( 0 \le x \le \pi \), the function \( f(x) = \cos x \) traces the typical cosine wave starting at 1 and ending at -1 within this interval.
• For \( x > \pi \), \( f(x) = \sin x \) continues the normal sine wave from where \( x = \pi \) onwards.

When sketching, it’s important to highlight points of transition, such as \( x = 0 \) and \( x = \pi \), to ensure clear understanding if the function remains continuous or exhibits a jump or other discontinuities. These sketches help in visualizing where and how the function might not be smooth or continuous, guiding further analysis in limit evaluation.

Limit Evaluation

Evaluating limits in piecewise functions involves examining the behavior of the function as it approaches specific points from either side. In this exercise, the critical points to evaluate are \( x = 0 \) and \( x = \pi \), where the function's expression changes:

• At \( x = 0 \), evaluate from the left using \( 1 + \sin x \) and from the right using \( \cos x \). Here, both paths approach the same value, 1, indicating a continuous function at this point.
• Conversely, at \( x = \pi \), evaluate from the left with \( \cos x \) (approaching -1) and from the right with \( \sin x \) (approaching 0). The discrepancy between these values suggests a discontinuity at \( x = \pi \).

For limits to exist at a specific point, the left-hand and right-hand limits must be equal. If they differ, as seen with \( x = \pi \), the limit doesn’t exist. Understanding these evaluations assists in determining where the function's behavior is smooth versus where it might break.

Trigonometric Functions

Trigonometric functions such as sine (\( \sin x \)) and cosine (\( \cos x \)) are pivotal in analyzing piecewise functions due to their periodic nature and well-known properties.

• The sine function, \( \sin x \), oscillates between -1 and 1 over its cycle, repeating every \( 2\pi \). In this exercise, for \( x > \pi \), the sine function gradually rises from 0 at \( x = \pi \) toward its peak.
• The cosine function, \( \cos x \), similarly cycles through values from 1 to -1 within its period. Here, it models values between \( 0 \) and \( \pi \), capturing a segment of its natural curve.

Both functions' behaviors at transitional points in the piecewise function influence the overall continuity and limit existence. A deep understanding of these functions aids not only in crafting accurate graphs but also in predicting where the piecewise function might experience continuity or discontinuity, crucial for correctly evaluating limits.