Problem 72 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(71 - 74 ,\) complete the following tables and state what you believe \(\lim _ { x \rightarrow 0 } f ( x )\) to be. $$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$ $$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\\ \hline \end{array}$$ $$f ( x ) = \sin \frac { 1 } { x }$$

Step-by-Step Solution

Answer

\(\lim _ { x \rightarrow 0 } f ( x )\) does not exist for the function \( f ( x ) = \sin \left( \frac { 1 } { x } \) \).

Step 1: Substitution from Negative Side

Evaluate \( f ( x ) = \sin \left( \frac { 1 } { x } \) \) for \( x \) values of -0.1, -0.01, -0.001, -0.0001. The corresponding \( f ( x ) \) values obtained by substituting these values into the function will show that \( f ( x ) \) keeps oscillating between -1 and 1 due to the sine function.

Step 2: Substitution from Positive Side

Evaluate \( f ( x ) = \sin \left( \frac { 1 } { x } \) \) for \( x \) values of 0.1, 0.01, 0.001, 0.0001. Similar to step 1, the resulting \( f ( x ) \) values will show an oscillating pattern between -1 and 1.

Step 3: Observation and Conclusion

Based on the results of Step 1 and Step 2, observing the behaviour of \( f ( x ) \) as \( x \) approaches 0 from both the negative and positive directions, it can be seen that the function does not approach a single number, rather it oscillates between -1 and 1. Thus, it is concluded that \(\lim _ { x \rightarrow 0 } f ( x )\) does not exist.

Key Concepts

Oscillating Functions

When we encounter functions like the sine and cosine, we're dealing with what are called oscillating functions. These functions exhibit repetitive behavior, similar to the swinging of a pendulum or the waves of the sea. They rise and fall in a consistent, predictable pattern. In calculus, understanding the behavior of oscillating functions, particularly as the input gets very small or very large, is crucial to mastering the concept of limits.

An excellent example of an oscillating function is given by the exercise with the function f(x) = sin(1/x). As x approaches zero, the value of 1/x increases without bound, causing the sine function to oscillate more and more rapidly. This is because, for very small values of x, even a tiny change in x results in a significant change in 1/x. Consequently, the sine function transitions through its cycle much faster, leading to oscillations that do not settle down to any specific value as x approaches zero.

Unbounded Behavior

The behavior of some functions can be described as unbounded when their values grow infinitely large or decrease infinitely without reaching a limit. In the case of oscillating functions like f(x) = sin(1/x), the term unbounded refers not to the function's values themselves, which for the sine function lie between -1 and 1, but to the input to the sine function, 1/x, as x approaches zero.

The behavior is unbounded in the sense that there's no limit to how large 1/x can grow, leading to more and more cycles of the sine function as x gets closer to zero. This characteristic shows why the limit of an oscillating function like this does not exist at certain points: the output values are not approaching any single number, but instead jumping back and forth between the extremes of the function's range.

Evaluating Limits Algebraically

In calculus, evaluating limits algebraically is a fundamental skill that involves manipulating expressions to find the value a function approaches as the input approaches a certain point. However, not all limits can be evaluated by simple algebraic manipulation, particularly when dealing with oscillating functions or those with unbounded behavior.

In the given exercise with f(x) = sin(1/x), attempting to evaluate the limit algebraically as x approaches zero can be misleading because the sine function does not cease its oscillations. Substituting values closer and closer to zero into x does not yield values that converge to a single point, as it might with polynomials or rational functions. Here, the behavior of the function must be observed and the limit must be evaluated based on the pattern of this behavior rather than by straightforward algebraic calculation. This example stresses the importance of coupling algebraic intuition with a deep understanding of a function's graph and behavior to accurately evaluate limits.