Question

Evaluate the given limits using the graph of the function. $$ \begin{array}{l} f(x)=\cos (x) \\ \text { (a) } \lim _{x \rightarrow-\infty} f(x) \\ \text { (b) } \lim _{x \rightarrow \infty} f(x) \end{array} $$

Short answer

Both limits do not exist due to the oscillatory nature of \( \cos(x) \).

Step-by-step solution

Understanding the Function

The function given is \( f(x) = \cos(x) \), which is a trigonometric function known as the cosine function. This function oscillates between -1 and 1 indefinitely.

Analyzing the Behavior as x Approaches Negative Infinity

As \( x \to -\infty \), the cosine function continues its periodic oscillation without settling at a particular value. The graph of \( \cos(x) \) does not approach a single value as \( x \to -\infty \).

Analyzing the Behavior as x Approaches Positive Infinity

Similarly, as \( x \to \infty \), the cosine function continues the same oscillation. The graph maintains its periodic nature and does not approach a single value as \( x \to \infty \).

Conclusion on Limits

Since the cosine function doesn't approach any single value as \( x \to \pm\infty \), both limits do not exist. The oscillatory nature of \( \cos(x) \) ensures it never settles.

Key concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often denoted as \( \cos(x) \). Its graph is a wave-like curve that repeats at regular intervals, known as a periodic function. Specifically, the cosine function repeats every \( 2\pi \) units along the x-axis, which is the period of the cosine graph.

Key characteristics of the cosine function include:

• The maximum value is 1, reached at points like \( x = 0, 2\pi, 4\pi, \ldots \)
• The minimum value is -1, reached at points like \( x = \pi, 3\pi, 5\pi, \ldots \)
• The function is even, meaning \( \cos(-x) = \cos(x) \).

Because \( \cos(x) \) doesn't rely on whether x is increasing or decreasing over an infinite scope, it does not stabilize or reach a finite limit as x heads towards infinity or negative infinity. The oscillating nature ensures that despite movement along the x-axis, the output continuously shifts between -1 and 1.

Infinite Limits

When we talk about infinite limits, we're referring to the behavior of functions as the input grows extremely large in the positive or negative direction. However, this concept doesn't necessarily mean the function itself heads towards infinity—instead, it's about what happens as input values move towards infinite boundaries.

For the cosine function \( \cos(x) \), determining the limit as \( x \to \infty \) or \( x \to -\infty \) involves observing the periodic and oscillatory nature. Because \( \cos(x) \) doesn't converge towards a specific value, it remains undefined or does not exist in these contexts.

This can feel counterintuitive, as many base functions either spike or stabilize at infinity but trigonometric functions, like the cosine, maintain a steady rhythmic pattern instead. Hence, when evaluating the limit of \( \cos(x) \) as \( x \to \pm \infty \), we conclude they "do not exist" due to lack of convergence to a particular value.

Graphical Analysis

Graphical analysis is a vital tool in understanding the behavior of functions, especially when dealing with complex limits. By plotting the cosine function, \( f(x) = \cos(x) \), we gain visual insight into its behavior across different ranges of x.

Using a graph:

• Observe that the plot exhibits a repeating wave pattern, oscillating between -1 and 1.
• Notice the absence of a trend towards a single value as x moves towards infinity in either direction. Instead, the graph shows consistent repetition.
• Recognize the periodicity from each complete wave cycle to the next, reaffirming no tendency to stabilize with increasing x.

Graphical analysis of \( \cos(x) \) reaffirms conclusions drawn from analytical approaches—such as limits being nonexistent due to constant oscillation in values.