Problem 26 - Precalculus: Mathematics for Calculus

Question

Estimating Limits Graphically Use a graphing device to determine whether the limit exists. If the limit exists, estimate its value to two decimal places. $$\lim _{x \rightarrow 0} \sin \frac{2}{x}$$

Step-by-Step Solution

Answer

The limit does not exist as the function oscillates rapidly near zero.

Step 1: Understanding the Problem

The task is to determine if the limit \( \lim_{x \to 0} \sin \frac{2}{x} \) exists and estimate its value graphically. We need a graphing tool to visualize this function near \( x = 0 \).

Step 2: Plotting the Function

Use a graphing device (like Desmos or a graphing calculator) to plot the function \( y = \sin \frac{2}{x} \). Focus on a small interval around \( x = 0 \), e.g., from \( -0.1 \) to \( 0.1 \).

Step 3: Analyzing the Graph

Observe the behavior of the graph as \( x \) approaches 0 from both sides. Check if the \( y \)-values are approaching a single value or not.

Step 4: Interpreting the Behavior

Notice that as \( x \) approaches 0, the function \( \sin \frac{2}{x} \) oscillates increasingly rapidly between -1 and 1 because the term \( \frac{2}{x} \) becomes very large in magnitude, causing rapid oscillation.

Step 5: Conclusion on Limit Existence

Since the function oscillates between -1 and 1 without approaching any particular value, the limit \( \lim_{x \to 0} \sin \frac{2}{x} \) does not exist.

Key Concepts

Limits of Functions

When we talk about the "limit of a function," we are exploring the behavior of a function as the input gets closer to a certain point. In mathematical terms, we're interested in what happens to the function's output as the input "approaches" a specific value.
For example, let's consider the function \( f(x) = \sin \frac{2}{x} \). We're trying to determine what happens to \( \sin \frac{2}{x} \) as \( x \) gets closer to 0. If the outputs are approaching a single, finite value, the limit exists.

• If the outputs converge to a particular number, the limit exists, and we can specify its value.
• If the outputs behave erratically or oscillate without settling, the limit does not exist.

In this case, \( \sin \frac{2}{x} \) becomes quite unpredictable near \( x = 0 \), oscillating rapidly. Thus, the limit does not exist because there's no single value the function is approaching.

Graphical Analysis

Graphical analysis is a powerful tool for understanding functions and their limits. By plotting the graph of a function, we can visually assess how the function behaves, especially as it approaches a point of interest.
For our example, plotting \( y = \sin \frac{2}{x} \) around \( x = 0 \) is crucial. When using a graphing calculator, focus on a narrow range, like from \(-0.1\) to \(0.1\), around \( x = 0 \).
Once plotted:

• Look for clear trends or patterns in the graph as \( x \) approaches 0 from both sides.
• Notice if the function values stabilize to approach a particular number, indicating that a limit might exist.
• If the graph shows erratic behavior or doesn't settle on one value, the limit likely does not exist.

In this instance, the graph of \( \sin \frac{2}{x} \) reveals rapid oscillations between -1 and 1 as \( x \) nears 0, indicating no stable value is approached. Thus, the graphical analysis confirms the limit does not exist.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are deeply connected to angles and periodic behavior. The sine function, \( \sin(x) \), is known for its smooth and periodic oscillations between -1 and 1.
For \( \sin \left(\frac{2}{x}\right) \), the argument of the sine function is transformed radically. As \( x \) moves closer to 0, \( \frac{2}{x} \) swings wildly due to the division by a small number, varying the input angle across a vast range instantly.
This results in:

• Rapid oscillation of the sine function between its extremes of -1 and 1.
• No distinct pattern or limit for \( \lim_{x \to 0} \sin \frac{2}{x} \) due to this unpredictable nature near the point.

Overall, understanding how trigonometric functions behave under different transformations is essential. In our particular example, the limit does not exist because the transformation causes the sine function to oscillate too rapidly for a singular limit value to develop.