Problem 58 - Precalculus

Question

Evaluate the following limits using a table of values. Given \(g(x)=\frac{\sin [\pi(x-1)]}{|x|},\) find a. $$\lim _{x \rightarrow 0^{-}} g(x)$$ b. $$\lim _{x \rightarrow 0^{+}} g(x)$$ c. $$\lim _{x \rightarrow 0} g(x)$$

Step-by-Step Solution

Answer

a. The limit is 0. b. The limit is 0. c. The limit exists and is 0.

Step 1: Understand the Function

We are given the function \(g(x) = \frac{\sin [\pi(x-1)]}{|x|}\). This function involves sine and an absolute value in the denominator, which makes it interesting around \(x = 0\) because of the potential for division by zero or undefined behavior. Our goal is to evaluate the limits at different approaches to zero.

Step 2: Evaluate \(\lim_{x \to 0^-} g(x)\) Using a Table

To find \(\lim_{x \to 0^-} g(x)\), we construct a table of values for \(g(x)\) as \(x\) approaches 0 from the negative side. Choose values like \(-0.1, -0.01, -0.001,\) and calculate \(g(x)\) for each. Remember, \(|x| = -x\) for negative \(x\). Compute the corresponding sine values and divide them by \(|x|\). Observe the trend as \(x\) gets closer to zero.For example:- \(x = -0.1,\, g(x) = \frac{\sin[\pi(-0.1 - 1)]}{0.1}\)- Continue with \(-0.01, -0.001,...\) and observe the pattern.

Step 3: Evaluate \(\lim_{x \to 0^+} g(x)\) Using a Table

Similarly, for \(\lim_{x \to 0^+} g(x)\), choose positive \(x\) values like \(0.1, 0.01, 0.001,\) etc. Here, \(|x| = x\) remains positive. Calculate \(g(x)\) for these points and observe its behavior as \(x\) nears zero from the right.For example:- \(x = 0.1,\, g(x) = \frac{\sin[\pi(0.1 - 1)]}{0.1}\)- Continue with \(0.01, 0.001,...\) and observe if there is a stable trend.

Step 4: Determine \(\lim_{x \to 0} g(x)\) Using One-Sided Limits

To find the two-sided limit \(\lim_{x \to 0} g(x)\), compare the one-sided limits \(\lim_{x \to 0^-} g(x)\) and \(\lim_{x \to 0^+} g(x)\). If these two are equal, then the two-sided limit exists and is equal to their common value. If they are different, the two-sided limit does not exist.

Step 5: Conclude the Limit Evaluations

Based on the calculations from Step 2 and Step 3, if both sides approach the same value, then the limit as \(x \to 0\) exists at that value. Otherwise, state that the limit does not exist. The calculations should show if \(g(x)\) tends towards zero or any other number symmetrically from both sides.

Key Concepts

Sine Function

The sine function, usually denoted as \(\sin(x)\), is a periodic function that forms part of the family of trigonometric functions. It's commonly used to model wave-like phenomena, whether sound or light waves. The wave-like nature of \(\sin(x)\) means it oscillates between -1 and 1.
A key aspect of the sine function is its periodicity:

• The function repeats every \(2\pi\) units.
• This characteristic pattern makes it predictable over its domain.

In the expression \(\pi(x-1)\) embedded within the sine function in our problem, the sine function is having its phase shifted. This phase shift influences where the peaks and troughs occur on the graph.
Understanding how this function behaves is essential when evaluating limits because it shows us how the sine values fluctuate as \(x\) changes.

Absolute Value

The absolute value of a number, denoted \(|x|\), describes the number's distance from zero on the number line, without considering its direction. Essentially, it's always a non-negative value.
Here are some properties:

• If \(x\) is positive or zero, \(|x| = x\).
• If \(x\) is negative, \(|x| = -x\).

In the given function \(g(x) = \frac{\sin [\pi(x-1)]}{|x|}\), the absolute value in the denominator ensures that the denominator is non-negative.
The absolute value plays a crucial role, especially near \(x = 0\), because it affects how the numerator is divided, impacting the behavior of the function close to zero.

One-Sided Limits

One-sided limits are limits that consider the behavior of a function as the input approaches a specific point from one side—either from the left (\(x \to c^-\)) or from the right (\(x \to c^+\)).
To evaluate one-sided limits:

• For \(\lim_{x \to c^-} f(x)\), we examine values of \(x\) approaching \(c\) from the negative side.
• For \(\lim_{x \to c^+} f(x)\), we consider values approaching from the positive side.

In this exercise, we used tables to compute values for \(g(x)\) as \(x\) approached 0 from negative and positive sides. By observing these values, we could determine the trend of the function as it neared zero from either direction. Such evaluations reveal whether the function becomes positive, negative, or approaches a specific number as \(x\) nears zero.

Two-Sided Limits

Two-sided limits involve considering the behavior of a function as the input approaches a specific point from both left and right. A two-sided limit exists at point \(c\) if both one-sided limits are equal.
Here's a step-by-step approach:

• Calculate \(\lim_{x \to c^-} f(x)\) by approaching from the left.
• Calculate \(\lim_{x \to c^+} f(x)\) by approaching from the right.
• Verify if the one-sided limits are equal.

In this problem, we checked both \(\lim_{x \to 0^-} g(x)\) and \(\lim_{x \to 0^+} g(x)\). By comparing these one-sided limits, we determined whether \(\lim_{x \to 0} g(x)\) exists as a common value. When both limits were the same, the two-sided limit existed, showing that the function has a consistent behavior around \(x = 0\). This helps identify points of continuity or discontinuity in functions involving complex expressions.