Problem 71 - Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The function is continuous on \((- fty, fty)\) as it is continuous everywhere in its domain.

Step 1: Understand the Definition of Continuity

The function \( f(x) \) is continuous at a point \( x = c \) if \( \lim_{x \to c} f(x) = f(c) \). We need to check this for the entire domain of \( f(x) \). This problem specifically requires verifying the continuity at \( x = 0 \) because \( f(x) \) is clearly continuous for \( x eq 0 \) since \( x^4 \sin (1/x) \) is a combination of continuous functions.

Step 2: Evaluate \( f(x) \) for \( x \neq 0 \)

For any \( x eq 0 \), the function \( f(x) = x^4 \sin (1/x) \) is continuous because it's made from \( x^4 \) and \( \sin(1/x) \), both of which are continuous. Therefore, \( f(x) \) is continuous for all non-zero values of \( x \).

Step 3: Evaluate Limit as \( x \to 0 \)

We need to check \( \lim_{x \to 0} x^4 \sin (1/x) \) to prove continuity at \( x = 0 \). Since \( \sin(1/x) \) is bounded by -1 and 1, we have \( -x^4 \leq x^4 \sin (1/x) \leq x^4 \). Taking limits gives \( \lim_{x \to 0} -x^4 = 0 \) and \( \lim_{x \to 0} x^4 = 0 \). Therefore, by the Squeeze Theorem, \( \lim_{x \to 0} x^4 \sin (1/x) = 0 \).

Step 4: Connect Continuity at \( x = 0 \)

We found that \( \lim_{x \to 0} f(x) = 0 \). Since \( f(0) = 0 \) by definition of the piecewise function, it follows that \( \lim_{x \to 0} f(x) = f(0) \). Thus, the function is continuous at \( x = 0 \).

Step 5: Conclusion of Continuity

Since we have shown that \( f(x) \) is continuous for \( x eq 0 \) and specifically at \( x = 0 \), we conclude that \( f(x) \) is continuous over the entire domain \( (-\infty, \infty) \).

Key Concepts

Squeeze Theorem

The Squeeze Theorem is a fascinating concept in calculus that helps us determine the limit of a function that is "sandwiched" between two other functions. Imagine you're squeezing a lemon; as you apply pressure from both sides, the juice is forced to move in a certain direction. This analogy helps in understanding how the Squeeze Theorem works: it forces the limit of a function to a specific value based on the behavior of its bounding functions.

For any given function \( f(x) \) sandwiched between two functions, say \( g(x) \) and \( h(x) \), if \( g(x) \leq f(x) \leq h(x) \) for all \( x \) in some interval around \( c \) (except possibly at \( c \) itself), and if \( \lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L \), then \( \lim_{x \to c} f(x) = L \) as well.

In the original exercise, we apply the Squeeze Theorem to the function \( g(x) = -x^4 \leq x^4 \sin (1/x) \leq h(x) = x^4 \). We know that both limits of \( -x^4 \) and \( x^4 \) as \( x \to 0 \) are \( 0 \). By the Squeeze Theorem, \( \lim_{x \to 0} x^4 \sin (1/x) = 0 \), squeezing the limit of \( f(x) \) to match this value.

Piecewise Function

A piecewise function is like a chameleon, changing its form depending on the input value. This clever type of function is defined by different expressions over distinct intervals of the domain. This means that the value or rule applied to calculate the function's output shifts at specified points.

In the given problem, the function \( f(x) \) helps create a piecewise scenario. It assigns \( f(x) = x^4 \sin(1/x) \) for \( x eq 0 \), and \( f(x) = 0 \) for \( x = 0 \). This definition allows the function to behave smoothly when \( x \) is not equal to zero, where both components \( x^4 \) and \( \sin(1/x) \) maintain continuity.

The beauty of piecewise functions is in their ability to model situations where a single equation isn't sufficient. They help bridge across boundaries where behavior "jumps," making them versatile for expressing complex and varied real-world phenomena.

Limit Evaluation

Evaluating limits involves closely examining the behavior of a function as it approaches a certain point. It's like slowly walking up to a doorstep and peering through the peephole to see how things appear right there at the threshold.

To evaluate limits, especially as in this problem at \( x = 0 \), we carefully consider \( \lim_{x \to 0} x^4 \sin(1/x) \). The function \( \sin(1/x) \) oscillates wildly, which might initially seem problematic. However, since it's multiplied by \( x^4 \), which shrinks to zero, the product also approaches zero.

Our limit evaluation becomes formal when using methods like substitution, factoring, or in this case, the Squeeze Theorem. With \( -x^4 \leq x^4 \sin(1/x) \leq x^4 \), both bounds approach zero as \( x \to 0 \), letting us conclude that \( \lim_{x \to 0} x^4 \sin(1/x) \) also equals zero. Limit evaluation taps into the fundamental aspects of calculus, providing insights into how functions behave in fine detail near specific points.