Problem 63 - Calculus Early Transcendentals

Question

Show that the function $$f(x)=\left\\{\begin{array}{ll}{x^{4} \sin (1 / x)} & {\text { if } x \neq 0} \\\ {0} & {\text { if } x=0}\end{array}\right.$$ is continuous on \((-\infty, \infty)\)

Step-by-Step Solution

Answer

The function \(f(x)\) is continuous on \((–\infty, \infty)\).

Step 1: Determine Definition and Points of Interest

First, identify that the function \(f(x)\) is piecewise defined. It is given by \(x^4 \sin(1/x)\) when \(x eq 0\) and 0 when \(x = 0\). We must determine continuity at all points, focusing particularly at \(x = 0\) since the expression changes at this point.

Step 2: Check Continuity for x ≠ 0

For any \(x eq 0\), the function \(f(x) = x^4 \sin(1/x)\) is a product of \(x^4\) and \(\sin(1/x)\). Both \(x^4\) and \(\sin(1/x)\) are continuous for \(x eq 0\), thus \(f(x)\) is continuous at all \(x eq 0\).

Step 3: Use Limit Definition of Continuity at x = 0

To check continuity at \(x = 0\), we need \(\lim_{x \to 0} f(x) = f(0) = 0\). Therefore, calculate \(\lim_{x \to 0} x^4 \sin(1/x)\).

Step 4: Determine Limit Behavior as x Approaches 0

Recall that \(-1 \leq \sin(1/x) \leq 1\). This bounds \(x^4 \sin(1/x)\) as \(-x^4 \leq x^4 \sin(1/x) \leq x^4\). Applying the Squeeze Theorem, since \(\lim_{x \to 0} -x^4 = 0\) and \(\lim_{x \to 0} x^4 = 0\), it follows that \(\lim_{x \to 0} x^4 \sin(1/x) = 0\).

Step 5: Conclude by Verifying Continuity at x = 0

As calculated, \(\lim_{x \to 0} f(x) = 0\) which equals \(f(0) = 0\). Thus, \(f(x)\) is continuous at \(x = 0\).

Step 6: Confirm Continuity Over Entire Domain

Having checked continuity both for \(x eq 0\) and at \(x = 0\), we conclude that the function \(f(x)\) is continuous for all \(x\).

Key Concepts

Piecewise Functions

Piecewise functions are unique in their structure as they are defined by different expressions for different intervals of the domain. In our problem, the function is given as:

• \( f(x) = x^4 \sin(1/x) \) for \( x eq 0 \)
• \( f(x) = 0 \) for \( x = 0 \)

This setup allows a function to behave differently based on which part of the domain is being considered. Understanding the behavior of these functions at the boundaries where the pieces meet or change is crucial for assessing the overall continuity of the function. Particularly, in this problem, we need to carefully examine the behavior at the point where the definition of the function changes, which is the point \( x = 0 \). A major part of dealing with piecewise functions is ensuring that the transition between 'pieces' of the function does not create discontinuities.

Squeeze Theorem

The Squeeze Theorem is a helpful tool in calculus used to find the limit of a function trapped between two other functions whose limits are known and equal at a point. To apply this in the context of our function, consider the range constraint on the sine function: \(-1 \leq \sin(1/x) \leq 1\).
The expression \( x^4 \sin(1/x) \) is naturally squeezed by these constraints because multiplying by \( x^4 \) scales the sine range:

• \( -x^4 \leq x^4 \sin(1/x) \leq x^4 \)

As \( x \to 0 \), both \( -x^4 \) and \( x^4 \) approach 0, thus according to the Squeeze Theorem, the limit of \( x^4 \sin(1/x) \) as \( x \to 0 \) is also 0. This conclusion is essential as it verifies the continuity of the function at the pivotal point \( x = 0 \). Squeezing thus enables us to conclude a limit behavior by understanding bounds, which can be challenging to compute directly otherwise.

Limit Definition of Continuity

The limit definition of continuity is foundational in understanding how a function behaves at a specific point. A function \( f(x) \) is continuous at a point \( c \) if:

• The limit \( \lim_{x \to c} f(x) \) exists.
• The function’s value exists at that point: \( f(c) \).
• Both the above are equal: \( \lim_{x \to c} f(x) = f(c) \).

For the function in this exercise, examining \( x = 0 \) is crucial since it is a transition point between two different definitions. First, we calculate the limit: \( \lim_{x \to 0} x^4 \sin(1/x) = 0 \) using the Squeeze Theorem.
This equals the directly defined function value at 0, \( f(0) = 0 \). Thus, the function is deemed continuous at \( x = 0 \). Ensuring that these conditions are satisfied across all points is how we determine the function's overall continuity in its domain.

Trigonometric Functions

Trigonometric functions, such as sine, play a significant role in various mathematical problems, including this exercise. The function \( \sin(1/x) \) is particularly intriguing because it involves division by \( x \), which can lead to rapid oscillations as \( x \) approaches zero. However, the behavior of \( \sin \) itself is well-bounded between -1 and 1. This bounded nature was leveraged in applying the Squeeze Theorem to establish limits. Understanding the inherent properties of trigonometric functions helps in analyzing complex expressions, such as those considered in piecewise definitions or wrapped within polynomial expressions. In this problem, by combining the oscillatory nature of the sine function with the smoothing effect of \( x^4 \), we explored how these properties contribute to analyzing continuity.