Problem 18 - Calculus for Biology and Medicine

Question

Let \(f(x)=x^{3} \cos \frac{1}{x}, \quad x \neq 0\) (a) Use a graphing calculator to sketch the graph of \(y=f(x)\). (b) Use the sandwich theorem to show that \(\lim _{x \rightarrow 0} x^{3} \cos \frac{1}{x}=0\).

Step-by-Step Solution

Answer

The limit is 0 by the sandwich theorem.

Step 1: Understanding the Function

We are dealing with the function \( f(x) = x^3 \cos \frac{1}{x} \) for \( x eq 0 \). This is a combination of a polynomial, \( x^3 \), and a trigonometric function, \( \cos \frac{1}{x} \). To analyze the limit at \( x = 0 \), it’s important to consider the behavior of each component as \( x \) approaches zero.

Step 2: Sketching the Graph

With a graphing calculator, input \( f(x) = x^3 \cos \frac{1}{x} \) and observe how the graph behaves as \( x \) approaches zero. You will notice that near zero, the graph oscillates rapidly due to \( \cos \) but is bounded and suppresses close to the origin due to \( x^3 \) approaching zero.

Step 3: Setting Up the Sandwich Theorem

The sandwich (squeeze) theorem states that if \( g(x) \leq f(x) \leq h(x) \) for all \( x \) in some interval around \( c \) (except possibly at \( c \)) and \( \lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L \), then \( \lim_{x \to c} f(x) = L \). In our case, \( f(x) = x^3 \cos \frac{1}{x} \), the function \( \cos \frac{1}{x} \) is always bounded between -1 and 1.

Step 4: Applying the Sandwich Theorem Bounds

For \( x^3 \cos \frac{1}{x} \), set the bounds using the inequality \(-1 \leq \cos \frac{1}{x} \leq 1\). Therefore, multiplying by \( x^3 \), we get \(-x^3 \leq x^3 \cos \frac{1}{x} \leq x^3\).

Step 5: Calculating the Limit of the Bounds

Calculate the limit of the lower and upper bounds as \( x \to 0 \). Since both \( \lim_{x \to 0} (-x^3) = 0 \) and \( \lim_{x \to 0} x^3 = 0 \), by the sandwich theorem, we conclude that \( \lim_{x \to 0} x^3 \cos \frac{1}{x} = 0 \).

Key Concepts

Understanding Function Behavior

The behavior of a function as it approaches a specific point can reveal much about its overall nature. In the case of the function \( f(x) = x^3 \cos \left( \frac{1}{x} \right) \), you are dealing with the combination of a polynomial \( x^3 \) and a trigonometric function \( \cos \left( \frac{1}{x} \right) \). As \( x \) draws closer to zero, the polynomial term \( x^3 \) tends towards zero. However, the \( \cos \left( \frac{1}{x} \right) \) term oscillates between -1 and 1 at an increasing frequency. This oscillation indicates that although the function remains bounded, it experiences rapid changes in value near zero. Understanding this behavior is crucial, as it serves as a foundation for analyzing and estimating the limit of the function as \( x \) approaches any particular value and is especially useful when combined with other methods, such as the Sandwich Theorem, to conclusively determine the limit.

Limit Calculation Using the Sandwich Theorem

The Sandwich Theorem, also known as the Squeeze Theorem, is a central concept in calculus used to calculate limits. It comes into play when you can 'sandwich' a complex function between two simpler functions with known limits. For the function \( f(x) = x^3 \cos \left( \frac{1}{x} \right) \), we use the property that the \( \cos \) term is always bounded between -1 and 1. By multiplying these bounds by \( x^3 \), we create two new functions: ie.,

• the lower bound \(-x^3\) and
• the upper bound \(x^3\).

As \( x \) approaches zero, both bounds tend to zero: \[ \lim_{x \to 0} -x^3 = 0 \quad \text{and} \quad \lim_{x \to 0} x^3 = 0 \]According to the Sandwich Theorem, since both side limits equal zero, the limit of the original \( f(x) \) function must also be zero:\[ \lim_{x \to 0} x^3 \cos \left( \frac{1}{x} \right) = 0 \]

Utilizing a Graphing Calculator

A graphing calculator is an essential tool for visualizing complex function behavior. In this exercise, it is invaluable for sketching \( f(x) = x^3 \cos \left( \frac{1}{x} \right) \). By inputting the function into the calculator, you can visually inspect its behavior as \( x \) approaches zero.The visualization will reveal rapid oscillations near zero, where the graph appears to vibrate around the origin but stay pressed close to the x-axis. This close proximity is due to the \( x^3 \) term, which drives the overall function value closer to zero despite the oscillations, effectively damping the effect of \( \cos \left( \frac{1}{x} \right) \). Such visuals aid in understanding how the function limits to zero with increasing precision and offer an intuitive insight into why methods like the Sandwich Theorem work so effectively in this context.

Exploring Trigonometric Functions

Trigonometric functions like cosine play a fundamental role in calculus due to their repetitive and periodic nature. In \( f(x) = x^3 \cos \left( \frac{1}{x} \right) \), \( \cos \left( \frac{1}{x} \right) \) oscillates with infinitely increasing frequency as \( x \) approaches zero. This frenetic oscillation makes direct computation of limits challenging without applying techniques like bounding or using specific theorems. Despite these challenges, the bounded nature of the cosine function is a saving grace. Since \( \cos \theta \) always lies between -1 and 1, it allows for the effect of \( \cos \left( \frac{1}{x} \right) \) to be tempered by other factors like \( x^3 \) when considering limits. This boundedness, combined with general properties of trigonometric functions, is pivotal in employing strategies such as the Sandwich Theorem to resolve limits effectively.