Problem 68 - A Graphical Approach to Precalculus with Limits

Question

Assume that \(f(x)\) has domain \([0, \infty)\). Find \(\lim f(x)\) if the graph of \(y=f(x)\) has oblique asymptote \(y=-2 x+3\)

Step-by-Step Solution

Answer

\(\lim_{x \to \infty} f(x) = -\infty\)

Step 1: Understanding Oblique Asymptote

An oblique asymptote indicates the behavior of a function as it approaches infinity. For a function \(f(x)\) with an oblique asymptote \(y = mx + b\), the function behaves similarly to the line \(y = mx + b\) as \(x\) tends to infinity.

Step 2: Identifying Asymptote Equation

The problem states that the oblique asymptote of \(f(x)\) is \(y = -2x + 3\). This means that as \(x\) increases without bound, \(f(x)\) approaches the line \(-2x + 3\).

Step 3: Applying Limit Definition for Oblique Asymptotes

The property of an oblique asymptote \(y = mx + b\) implies that \(\lim_{x \to \infty}(f(x) - (-2x + 3)) = 0\). This means that the difference between \(f(x)\) and \(-2x + 3\) approaches zero as \(x\) tends to infinity.

Step 4: Deriving the Limit Expression

Given the asymptote equation, we can conclude that as \(x\) approaches infinity, \(f(x)\) follows the line closely. Therefore, \(\lim_{x \to \infty} f(x) = \lim_{x \to \infty}(-2x + 3)\).

Step 5: Evaluating the Limit

Since \(-2x\) approaches \(-\infty\) and the constant term \(3\) doesn't affect the infinity value, \(\lim_{x \to \infty}(-2x + 3) = -\infty\). Hence, \(\lim_{x \to \infty} f(x) = -\infty\).

Key Concepts

Limits

When talking about limits, we refer to the behavior of a function as it approaches a particular point or value. In mathematical terms, the limit tells us what value the function is approaching as the independent variable gets closer to some number.

For example, the limit \(\lim_{x \to \infty} f(x)\) describes the behavior of the function \(f(x)\) as \(x\) becomes infinitely large. In the context of our exercise, we're interested in how \(f(x)\) behaves as \(x\) grows towards infinity. An important note is that a limit can result in a finite value, infinity, or may not exist.

In our particular case, considering the asymptotic behavior of \(f(x)\) with its oblique asymptote \(y = -2x + 3\), we find that \(\lim_{x \to \infty} f(x) = -\infty\). This tells us that as \(x\) goes to infinity, \(f(x)\) keeps decreasing indefinitely.

Infinite Behavior

Infinite behavior in functions refers to what happens as the input values, typically represented by \(x\), grow either very large or very small. When studying this behavior, it helps us understand how a function behaves in the extremes, which sometimes can create interesting patterns like lines or curves, often called asymptotes.

Here, we're looking at the function \(f(x)\) that has an oblique asymptote of \(y = -2x + 3\). This means that while \(f(x)\) doesn't actually become the line \(-2x + 3\), it gets very close to it when \(x\) becomes very large. In other words, the difference between \(f(x)\) and \(-2x + 3\) diminishes as \(x\) heads towards infinity.

By understanding this behavior, we're able to conclude that even though \(f(x)\) continues to decrease, it moves closely alongside the line \(-2x + 3\) infinitely.

Function Behavior

Function behavior encompasses how a function behaves or changes based on the input values, providing a deeper understanding of its general behavior, such as trends, extremes, and asymptotic actions.

With respect to the oblique asymptote of \(y = -2x + 3\), it indicates that as \(f(x)\) approaches this line quickly when \(x\) increases without bounds. The oblique nature means the slope isn't zero, and thus the function linearly correlates with \(x\) with an added constant. This points out that even if \(f(x)\) might have bumps or wiggles, asymptotically it'll follow that slanting path.

Analyzing functions in this manner helps us predict and sketch where the function might lean as inputs get particularly large, offering insight into the potential range and reach of a function. By seeing how \(f(x)\)'s graph hugs the asymptote, we're better equipped to understand its dynamics and anticipate values over a large domain.