Problem 59 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

The accompanying graph represents a function \(g(x)\) that oscillates more and more frequently as approaches 0 from either the right or the left but with decreasing magnitude. Does \(\lim _{x \rightarrow 0} g(x)\) exist? If so, what is its value? [Note: For students with experience in trigonometry, the function \(g(x)=|x| \sin (1 / x)\) behaves in this way.]

Step-by-Step Solution

Answer

\( \lim_{{x \to 0}} g(x) = 0 \)

Step 1: Understand the behavior of the function

Analyze the given function or the description of the graph. The function oscillates frequently as it approaches 0 from either side. This means that as x approaches 0, the values of the function change rapidly.

Step 2: Consider the magnitude of oscillations

Note that the oscillations occur with decreasing magnitude. Therefore, even though the function changes rapidly, the values of these changes become smaller as x approaches 0.

Step 3: Define the limits approaching from the left and the right

Check the limit of the function as x approaches 0 from the left and the right. For the function described as \(g(x) = |x| \, \text{sin}(1/x)\), the important observation is that \(|x|\) tends to 0.

Step 4: Evaluate the limit of oscillations product

The \(\text{sin}(1/x)\) oscillates between -1 and 1, but since the preceding term is \(|x|\), the resulting product \(|x| \sin(1/x)\) squeezes the function value towards 0.

Step 5: Conclude the existence of the limit

Thus, since the magnitude of oscillations becomes insignificant as x approaches 0, the overall limit value converges to 0.

Key Concepts

Oscillating functions

Oscillating functions are functions that switch back and forth between values as they progress. These oscillations can be fast or slow. An example of oscillating functions in trigonometry is the sine or cosine functions. In the exercise above, the function \[ g(x) = |x| \, \text{sin}(1/x) \] oscillates. As x approaches zero from either side, it causes the sine term to change very quickly. However, just the presence of oscillations is not enough. We also need to know their behavior near 0.

Magnitude of oscillations

The magnitude of oscillations tells us how far the function values move from the central point during oscillations. For example, for the function \[ g(x) = |x| \, \text{sin}(1/x) \] as x gets closer to 0, the term |x| also gets closer to 0. Therefore, even though the sine function oscillates between -1 and 1, multiplying by a shrinking |x| results in the oscillations becoming smaller in magnitude. So, while the values of \(g(x)\) change rapidly, they do so within a range that continually decreases as x approaches 0.

Approaching zero

When examining the limit of a function as x approaches a certain value, like 0, it's crucial to understand how each component of the function behaves. For the function in question, \[ g(x) = |x| \, \text{sin}(1/x) \] the magnitude of \[ |x| \] gets increasingly smaller as x approaches 0. The sine function, \( \text{sin}(1/x) \), continues to oscillate rapidly between -1 and 1, but since the product includes |x| which approaches 0, the overall effect is that the values of \(g(x)\) also approach 0. In limit terms, despite the oscillations, \[ \lim _{x \rightarrow 0} g(x) = 0 \]

Trigonometry-based functions

Trigonometry-based functions, such as sine and cosine, are often involved in problems about oscillations. The sine function, \(\text{sin}(1/x)\), is particularly interesting since its period gets smaller as 1/x increases, leading to rapid oscillations. However, when combined with a function that approaches 0, like |x| in the provided problem, these oscillations result in values that increasingly approach 0. This combination is a common technique in calculus to demonstrate limits of products of oscillating and shrinking quantities.