Question

, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 0} \cos (1 / x) $$

Short answer

The limit does not exist.

Step-by-step solution

Understand the Problem

We need to find the limit of the function \( \cos(1/x) \) as \( x \) approaches 0. In mathematical terms, we express this as \( \lim_{x \to 0} \cos(1/x) \).

Analyze the Behavior Near Zero

As \( x \) approaches 0, \( 1/x \) becomes very large, swinging from positive to negative values depending on the direction from which \( x \) approaches 0. This means \( \cos(1/x) \) oscillates increasingly rapidly between -1 and 1.

Consider Oscillating Function

Since \( \cos(1/x) \) does not settle to any particular value but instead oscillates between -1 and 1 without stabilization as \( x \) approaches 0, it becomes clear that the limit does not converge to any specific number.

Conclude About the Limit

Due to the rapid oscillation and lack of convergence to a particular value, we conclude that the limit \( \lim_{x \to 0} \cos(1/x) \) does not exist.

Key concepts

Oscillating Functions

Oscillating functions are a fascinating area in calculus, especially when discussing limits. These functions do not stabilize at a single value as input approaches a particular point. Instead, they swing back and forth between different values. For example, consider the function \[\cos(1/x)\] as \(x\) approaches 0. When \(x\) is positive and gets closer to zero, \(1/x\) increases sharply. As \(x\) becomes negative, \(1/x\) also increases rapidly, but negatively. This causes the cosine function to oscillate wildly between -1 and 1.
The key characteristic of oscillating functions is their inability to settle. This continuous back-and-forth nature can make finding limits challenging. Oscillating functions typically fluctuate infinitely without adhering to one value near certain points like zero. When analyzing such functions, it's crucial to observe this behavior to understand why limits may not exist.

Trigonometric Limits

Trigonometric limits often come into play when dealing with oscillating functions. These types of limits involve functions defined by trigonometric terms such as sine, cosine, or tangent, where behavior near critical points like zero is crucial. Let's take a closer look at \[\cos(1/x)\] as \(x\) converges towards zero.
In trigonometric functions, particularly cosine and sine, the periodicity is a critical trait. Cosine, for instance, has a period of \(2\pi\), meaning it completes a cycle every \(2\pi\) units of length. When we substitute something like \(1/x\) into the cosine function, the rapid change in \(1/x\) results in extremely rapid cycling of cosine waves.
Recognizing these swift oscillations in trig-based functions enables us to determine behavior near essential points. For the limit \(\lim _{x \rightarrow 0} \cos(1/x)\), the cycling does not settle, reflecting the typical pattern of trigonometric limits under certain transformations like \(1/x\).

• Trigonometric limits require an understanding of periodic nature.


• Changes in input terms such as \(1/x\) magnify oscillations.


• This causes functions to behave unexpectedly near critical points.

Non-Existent Limits

The concept of non-existent limits is fundamental in calculus, often emerging with oscillating functions or undefined points. When we approach the limit \[\lim _{x \rightarrow 0} \cos(1/x)\], it becomes evident that the result is unpredictable, fluctuating, and non-convergent. To understand why limits may not exist, it is essential to analyze their failure to settle on one value.
Non-existent limits occur when a function does not stabilize near a point but continues oscillating. The case \(\cos(1/x)\) clearly illustrates this, as it swings infinitely between -1 and 1 without nearing any definitive number.
Different types of non-existent limits arise under certain conditions:

• If the function fails to get closer to a specific numerical value.


• When the function's oscillation becomes more frequent as it nears a point.


• If multiple directional approaches lead to different values, implying divergence.

Understanding non-existent limits is crucial while studying calculus, especially as it broadens the comprehension of functions and continuity (or lack thereof) at different points.