Problem 1 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(1-8,\) use graphs and tables to find (a) \(\lim _{x \rightarrow \infty} f(x)\) and (b) \(\lim _{x \rightarrow-\infty} f(x)\) (c) Identify all horizontal asymptotes. $$f(x)=\cos \left(\frac{1}{x}\right)$$

Step-by-Step Solution

Answer

The limits as \(x\) approaches infinity and negative infinity do not exist as the function oscillates between -1 and 1. The horizontal asymptotes for the function are \(y = 1\) and \(y = -1\).

Step 1: Understanding the behaviour of the function

The function \(f(x)=\cos \left(\frac{1}{x}\right)\) becomes more oscillatory as \(x\) approaches 0. Cosine function oscillates between -1 and 1, hence for large \(x\), the value of \(f(x)\) will oscillate between -1 and 1.

Step 2: Finding the limit as \(x\) approaches infinity

As \(x\) becomes larger, \(\frac{1}{x}\) becomes closer to 0. Hence, the function tends to oscillate faster and faster. Thus, the limit as \(x\) approaches infinity is undefined because the function does not approach a particular value but oscillates between -1 and 1.

Step 3: Finding the limit as \(x\) approaches negative infinity

Similarly, as \(x\) becomes infinitely negative, our function also oscillates between -1 and 1 without settling down to a particular number. So the limit as \(x\) approaches negative infinity is also undefined.

Step 4: Identify horizontal asymptotes

A horizontal asymptote is a horizontal line that the curve approaches as \(x\) tends to infinity or negative infinity. Since function \(f(x)\) oscillates between -1 and 1, the lines \(y = 1\) and \(y = -1\) are the horizontal asymptotes.

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes are flat lines that a function approaches as the input variable, usually named \(x\), heads off to infinity or negative infinity. They're great for understanding the long-term behavior of a function. In our exercise, the function \(f(x) = \cos\left(\frac{1}{x}\right)\) has horizontal asymptotes. Even though \(f(x)\) doesn't settle at a single value as \(x\) heads towards infinity, it does hover between two boundaries. This suggests the presence of horizontal asymptotes.

• For \(f(x) = \cos\left(\frac{1}{x}\right)\), the cosine part ranges between -1 and 1.
• As \(x\) becomes very large or very small, \(\frac{1}{x}\) gets closer to zero, making the function \(\cos\left(\frac{1}{x}\right)\) oscillate between -1 and 1.
• Thus, the horizontal asymptotes are \(y = -1\) and \(y = 1\).

By understanding these boundaries, we can infer how the function behaves in the larger picture, though it never actually "hits" these asymptotes.

Infinite Limits

Infinite limits involve the behavior of a function as \(x\) approaches infinity or negative infinity. It's like asking, "What happens to the function as we go really far along the \(x\)-axis?" In the context of our function \(f(x) = \cos\left(\frac{1}{x}\right)\), things get interesting.

• As \(x\) tends to infinity, \(\frac{1}{x}\) shrinks towards zero. But, since cosine oscillates, \(f(x)\) continually shifts between -1 and 1.
• Unlike many functions which may approach a specific number, \(f(x)\) doesn't settle at a constant value.
• This means that the limits – \(\lim_{x \to \infty} f(x)\) and \(\lim_{x \to -\infty} f(x)\) – are both undefined.

Interestingly, although the infinite limits do not exist, the concept of oscillation provides crucial insights into the function's behavior as \(x\) progresses indefinitely.

Oscillating Functions

Oscillating functions are those that move back and forth between certain bounds without settling on a particular value. The function \(f(x)=\cos\left(\frac{1}{x}\right)\) is a perfect example. It demonstrates fascinating traits as \(x\) approaches both positive and negative extremes

• The term \(\frac{1}{x}\) means that as \(x\) grows, its reciprocal shrinks toward zero, causing \(\cos\left(\frac{1}{x}\right)\) to cycle rapidly between -1 and 1.
• This cyclic nature implies no steady limit at infinity or negative infinity, yet hints at a structured pattern.
• Despite its perpetual oscillation, the function stays confined within fixed bounds: -1 and 1.

These fluctuations make oscillating functions unique, often leading to undefined limits at infinity due to their relentless back-and-forth movement. They don't settle down, but instead dance within their range eternally.