91Ó°ÊÓ

The concept of the limit of a function is crucial in understanding the behavior of functions as the input approaches a particular value, such as infinity. In this case, we're looking at what happens to the function \(f(x)\) as \(x\) moves towards infinity, specifically within the interval \([1, +\infty)\). When dealing with limits, we're interested in the tendency or direction a function's value switches to, rather than pinpointing an exact value. For the given exercise, we're tasked to determine the limit of \(f(x)\) based on the bounds provided by the inequalities \(-\frac{1}{x} \leq f(x) \leq \frac{1}{x} \). As \(x\) becomes larger and larger, both \(-\frac{1}{x}\) and \(\frac{1}{x}\) trend towards zero. With \(f(x)\) squeezed between these two functions, the Squeeze Theorem can be applied to confirm that \( \lim_{{x \to \infty}} f(x) = 0 \). This theorem is particularly useful because it provides a definitive endpoint for \(f(x)\) without needing to explicitly calculate \(f(x)\)'s form. Therefore, understanding this aspect is essential for using the Squeeze Theorem to its fullest potential whenever a function is trapped between two bounding trends.

Asymptotic behavior describes how functions behave as the input variables grow infinitely large. It helps us predict the future behavior of a function beyond the scope of simple calculations. Understanding this concept is vital for sketching and analyzing functions. For the functions \(y=\frac{1}{x}\) and \(y=-\frac{1}{x}\), their asymptotic behavior is evident as \(x\) approaches infinity. Both functions approach the x-axis without touching it, meaning their y-values trend towards zero asymptotically. This gives them the characteristic of approaching a boundary—a line that the function comes infinitely close to but never intersects. This often refers to the horizontal asymptote in the context of graph sketching.For a function \(f(x)\) that fits between these two curves, \(-\frac{1}{x}\) and \(\frac{1}{x}\), its asymptotic behavior is likewise constrained. This squeezing action confines \(f(x)\) ever more closely to zero, influencing its shape and limiting behavior as \(x\) increases. Recognizing that both upper and lower bounds asymptotically move toward a common axis enables us to predict that \(f(x)\), too, will trend towards this value as \(x\) expands indefinitely.

Graph sketching is an invaluable skill in mathematics, helping visually interpret the behavior of functions. The task in this exercise focused on sketching the graphs of \(y=\frac{1}{x}\), \(y=-\frac{1}{x}\), and a function \(f(x)\) that sits between them.To effectively sketch these graphs:

• First outline \(y=\frac{1}{x}\). This curve decreases from positive one towards zero in the first quadrant, moving closer to the x-axis.
• Similarly, \(y=-\frac{1}{x}\) begins at negative one and increases towards zero in the fourth quadrant, keeping underneath the x-axis.
• Draw these on the same coordinate plane. Ensure they are visibly approaching the x-axis as they extend to the right.

For \(f(x)\) that adheres to the inequality \(-\frac{1}{x} \leq f(x) \leq \frac{1}{x}\), sketching requires maintaining it within these bounds. There is no specific curve defined for \(f(x)\) because it can take multiple forms respecting the inequality. What matters is the constraint, keeping it sashaying between the \(-\frac{1}{x}\) and \(\frac{1}{x}\) curves, moving ever closer to zero. This visual representation helps clarify how \(f(x)\) behaves across \([1, +\infty)\), squeezing towards the horizontal asymptote, the x-axis.