Problem 16 - APEX Calculus

Question

Evaluate the given limits of the piecewise defined functions \(f\). $$ \begin{array}{ll} f(x)=\left\\{\begin{array}{ll} \cos x & x<\pi \\ \sin x & x \geq \pi \end{array}\right. & \\ \begin{array}{ll} \text { (a) } \lim _{x \rightarrow \pi^{-}} f(x) & \text { (c) } \lim _{x \rightarrow \pi} f(x) \\ \text { (b) } \lim _{x \rightarrow \pi^{+}} f(x) & & \text { (d) } f(\pi) \end{array} \end{array} $$

Step-by-Step Solution

Answer

(a) -1, (b) 0, (c) Does not exist, (d) 0.

Step 1: Evaluate the left-hand limit at \( x = \pi \)

The left-hand limit as \( x \) approaches \( \pi \) from the left, denoted as \( \lim_{x \to \pi^-} f(x) \), considers values of \( x \) that are slightly less than \( \pi \). For these values, based on the piecewise function definition, \( f(x) = \cos x \). Thus, \( \lim_{x \to \pi^-} f(x) = \cos \pi = -1 \).

Step 2: Evaluate the right-hand limit at \( x = \pi \)

The right-hand limit as \( x \) approaches \( \pi \) from the right, denoted as \( \lim_{x \to \pi^+} f(x) \), considers values of \( x \) that are slightly greater than \( \pi \). For these values, based on the piecewise function definition, \( f(x) = \sin x \). Therefore, \( \lim_{x \to \pi^+} f(x) = \sin \pi = 0 \).

Step 3: Determine the overall limit at \( x = \pi \)

The overall limit \( \lim_{x \to \pi} f(x) \) exists if both the left-hand limit and the right-hand limit at \( x = \pi \) are equal. Since \( \lim_{x \to \pi^-} f(x) = -1 \) and \( \lim_{x \to \pi^+} f(x) = 0 \), they are not equal. Therefore, \( \lim_{x \to \pi} f(x) \) does not exist.

Step 4: Evaluate \( f(\pi) \)

To find \( f(\pi) \), substitute \( x = \pi \) into the piecewise definition. Since \( x = \pi \) falls into the condition \( x \geq \pi \), we use \( f(x) = \sin x \). Thus, \( f(\pi) = \sin \pi = 0 \).

Key Concepts

Piecewise Functions

Piecewise functions are special types of mathematical functions that have different expressions for different intervals of a variable. These functions are crucial because they help us model real-world situations where a rule or a pattern changes based on the input value.

In a piecewise function, each piece or segment is defined by a condition. For example, in the function given in the exercise, we have:

• For values of \( x < \pi \), the function is defined as \( f(x) = \cos x \).
• For values of \( x \geq \pi \), the function is defined as \( f(x) = \sin x \).

This means that the function behaves differently depending on whether the input \( x \) is less than or greater than \( \pi \). Understanding piecewise functions is essential for analyzing situations where behavior changes over intervals.

Left-Hand Limit

The left-hand limit of a function at a particular point refers to the value that the function approaches as its input gets close to that point from the left side, or from smaller values.

In our exercise, the left-hand limit as \( x \) approaches \( \pi \) from the left, denoted as \( \lim_{x \to \pi^-} f(x) \), is concerned with the behavior of the function as \( x \) gets closer to \( \pi \) but never actually reaching it. For \( x < \pi \), the function adheres to \( f(x) = \cos x \), which translates to computing \( \cos \pi = -1 \) when approached from the left side.

This concept helps us determine whether the function is continuous or if any discontinuities occur at specific points.

Right-Hand Limit

Right-hand limits describe the value a function approaches as the input nears a specific point from the right side, or from larger values. This is crucial for understanding how a function behaves coming from greater values towards a particular boundary.

In the given piecewise function, the right-hand limit as \( x \) approaches \( \pi \), denoted as \( \lim_{x \to \pi^+} f(x) \), focuses on inputs greater than \( \pi \). From the definition, when \( x \geq \pi \), the function becomes \( f(x) = \sin x \). Evaluating this right-hand limit gives us \( \sin \pi = 0 \).

The right-hand limit is a powerful tool in determining whether a function behaves uniformly from both sides of a specific point.

Trigonometric Functions

Trigonometric functions, like sin and cos, are essential to mathematics, especially in the study of periodic phenomena and circle-related problems. These functions are well-defined and periodic, with specific values at key points.

In the exercise, we utilize two primary trigonometric functions: cosine and sine. Here’s a quick overview:

• \( \cos x \): This function gives the x-coordinate of a point on the unit circle for a given angle. Notably, \( \cos \pi = -1 \).
• \( \sin x \): This function returns the y-coordinate of a point on the unit circle. For \( \pi \), \( \sin \pi = 0 \).

These values are essential for solving the problem, as they are used to determine the limits and continuity in the piecewise function scenario shown in the exercise. Understanding trigonometric functions helps us analyze cyclic patterns and oscillations effectively.