Problem 47 - Precalculus

Question

Evaluate the following limits using a table of values. Given \(f(x)=\frac{\sin x}{|\sin x|},\) find a. $$\lim _{x \rightarrow \pi} f(x)$$ b. $$\lim _{x \rightarrow \pi^{+}} f(x)$$

Step-by-Step Solution

Answer

a. The limit does not exist, b. The right-hand limit is 1.

Step 1: Understand the Function

The function given is \( f(x) = \frac{\sin x}{|\sin x|} \). This function evaluates to 1 when \( \sin x > 0 \), -1 when \( \sin x < 0 \), and is undefined when \( \sin x = 0 \).

Step 2: Evaluate Limit as x Approaches π from Both Sides

Since \( \sin x = 0 \) at \( x = \pi \), the function is undefined at this point. We need to analyze the behavior of \( f(x) \) as \( x \) approaches \( \pi \) from the left and the right to understand the limit.

Step 3: Evaluate Left-Hand Limit as x Approaches π

For \( x \) just slightly less than \( \pi \) (i.e., \( x \rightarrow \pi^- \)), \( \sin x \) is negative. Hence, \( f(x) = \frac{\sin x}{|\sin x|} = \frac{\text{negative}}{\text{positive}} = -1 \).

Step 4: Evaluate Right-Hand Limit as x Approaches π

For \( x \) just slightly greater than \( \pi \) (i.e., \( x \rightarrow \pi^+ \)), \( \sin x \) is positive. Hence, \( f(x) = \frac{\sin x}{|\sin x|} = \frac{\text{positive}}{\text{positive}} = 1 \).

Step 5: Conclude the Two-Sided Limit at π

Since the left-hand limit (\( x \rightarrow \pi^- \)) is -1 and the right-hand limit (\( x \rightarrow \pi^+ \)) is 1, the overall limit \( \lim_{x \rightarrow \pi} f(x) \) does not exist due to the mismatch.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), are fundamental in mathematics. They relate angles to ratios of sides in a right triangle. The sine function specifically generates values based on the y-coordinates of points on a unit circle as the angle increases.
In this exercise, we focus on the sine function. The function given, \( f(x) = \frac{\sin x}{|\sin x|} \), involves a sine function in the numerator and its absolute value in the denominator. This means the function's behavior depends on whether \( \sin x \) is positive, negative, or zero. Each trigonometric function has unique properties and periodicity, which are crucial for analyzing limits and function behavior.

Left-Hand Limit

The left-hand limit is the value that a function approaches as the input approaches from the left side of a specific point (for example, coming from slightly smaller values than \( \pi \)). Mathematically, it is expressed as \( \lim_{x \to c^-} f(x) \).
In this exercise, when we look at \( x \) approaching \( \pi \) from the left, specifically \( x \to \pi^- \), the sine of values just smaller than \( \pi \) is negative. Therefore, \( f(x) \) evaluated slightly less than \( \pi \) would be \( \frac{\text{negative}}{\text{positive}} = -1 \). This means the left-hand limit for \( x \) approaching \( \pi \) is -1. Understanding left-hand limits helps to predict how functions behave just before reaching specific points or values.

Right-Hand Limit

The right-hand limit is the value a function approaches as the input comes from the right side of a certain point. It is mathematically expressed as \( \lim_{x \to c^+} f(x) \).
For this exercise, as \( x \) approaches \( \pi \) from the right (\( x \to \pi^+ \)), the sine of values just larger than \( \pi \) becomes positive. Therefore, \( f(x) = \frac{\text{positive}}{\text{positive}} = 1 \). This indicates that the right-hand limit as \( x \) approaches \( \pi \) is \( 1 \). Right-hand limits are important for understanding how functions behave just after crossing certain points and comparing these with left-hand limits assures comprehensive analysis.

Undefined Function

A function becomes undefined at a specific point when it outputs fractions with zero in the denominator, or when some operations do not produce real numbers. In the context of limits, understanding when a function is undefined helps identify breaks or discontinuities.
In this case, at \( x = \pi \), \( \sin x = 0 \), leading to \( \frac{0}{0} \), which is undefined. That's why \( f(x) \) is not defined exactly at \( \pi \). Although the function is undefined at this point, exploring left-hand and right-hand limits offers insights into the behavior of functions around undefined areas. This is key when determining continuity and analyzing the overall behavior of functions around critical points.