Problem 33 - Precalculus with Limits

Question

In Exercises 29-34, use a graphing utility to graph the function and verify that the horizontal asymptote corresponds to the limit at infinity. $$ y = 1-\dfrac{3}{x^2} $$

Step-by-Step Solution

Answer

The graph of the function \(y = 1-\frac{3}{x^2}\) approaches a horizontal line as \(x\) approaches infinity or negative infinity. The horizontal asymptote of this function is \(y=1\), which is validated by the computed limits at infinity.

Step 1: Graph the Function

Graph the function \(y = 1-\frac{3}{x^2}\) using a graphing utility. Observe the behavior of the graph as \(x\) increases towards positive or negative infinity. The graph will approach a certain line without ever reaching or crossing it, which defines the horizontal asymptote.

Step 2: Compute the Limit at Infinity

To confirm that the horizontal asymptote is accurately reflected in the graph, the limit of the function as \(x\) approaches infinity or negative infinity must be calculated. As \(x\) increases or decreases without bound, any terms with \(x\) in the denominator should essentially become zero. Hence, calculate \(lim_{x\to+\infty}(1-\frac{3}{x^2}) = 1-0 = 1\) and \(lim_{x\to-\infty}(1-\frac{3}{x^2}) = 1-0 = 1\).

Step 3: Assess the Horizontal Asymptote

The line that the function approaches as \(x\) approaches infinity or negative infinity is represented by the equation \(y = 1\). This is confirmed by our computation of the limit, thereby verifying its position as the horizontal asymptote.

Key Concepts

limits at infinity

When we talk about "limits at infinity," we are discussing how a function behaves as the input values, denoted as \( x \), become extremely large (positive infinity) or extremely small (negative infinity). In mathematical terms, we're looking at what value the function approaches as \( x \to +\) or \( x \to -\).For the function \( y = 1 - \frac{3}{x^2} \), we notice a unique behavior as \( x \) becomes very large or very small. The fraction \( \frac{3}{x^2} \) contains \( x^2 \) in the denominator, which means as \( x \) increases or decreases significantly, the term \( \frac{3}{x^2} \) becomes negligibly small.

• As \( x \to +\): the fraction \( \frac{3}{x^2} \) approaches 0, and thus \( y \to 1 - 0 = 1 \).
• As \( x \to -\): again, the fraction \( \frac{3}{x^2} \) approaches 0, leading \( y \to 1 - 0 = 1 \).

Therefore, the limits help us find that the function is approaching \( y = 1 \) as \( x \to +\infty \) or \( x \to -\infty \).

graphing utility

Graphing utilities are tools that help us visualize mathematical functions, making it easier to see trends and behaviors that would be complex to calculate manually. These utilities can be software-based, like graphing calculators or computer programs.When graphing the function \( y = 1 - \frac{3}{x^2} \) using a graphing utility:

• You can observe how the graph behaves as \( x \) moves towards very large positive and negative values.
• The graph should clearly show the curve approaching an invisible horizontal line at \( y = 1 \) without touching or crossing it.
• This visual help allows for a better understanding of the concept of horizontal asymptotes.

Graphing utilities, therefore, provide a tangible representation of abstract mathematical concepts, making understanding both intuitive and insightful.

function behavior

Understanding function behavior involves studying how functions react at different points or intervals, essentially describing what functions "do" as their variables change. This behavior is often linked to limits and asymptotes. For the function \( y = 1 - \frac{3}{x^2} \), its behavior is affected by the term \( \frac{3}{x^2} \).Let's break it down:

• As \( x \to +\): The function gets closer and closer to the value 1, since \( \frac{3}{x^2} \) reduces towards zero.
• As \( x \to -\): Similarly, \( y \to 1 \) for the same reason: the diminishing effect of \( \frac{3}{x^2} \).
• This repetitive approach towards the value 1 as \( x \) tends towards infinity, in either direction, showcases that the function stabilizes or "settles" against the line \( y = 1 \), indicating a horizontal asymptote here.

This information is crucial for sketching accurate graphs and predicting function behaviors in real-world applications or higher-level mathematics.