Problem 9 - Precalculus Mathematics for Calculus

Question

Find the limit. $$\lim _{x \rightarrow-\infty} \frac{4 x^{2}+1}{2+3 x^{2}}$$

Step-by-Step Solution

Answer

As \(x\) approaches \(-\infty\), the limit is \(\frac{4}{3}\).

Step 1: Identify the Highest Power in the Denominator

The first step in finding the limit as \(x\) approaches \(-\infty\) is to identify the term in the denominator with the highest power in \(x\). In this case, the highest power of \(x\) in the denominator \(2 + 3x^2\) is \(x^2\).

Step 2: Simplify the Expression

To simplify, divide every term in the numerator and the denominator by \(x^2\), the highest power in the denominator. The expression becomes:\[\frac{4x^2/x^2 + 1/x^2}{2/x^2 + 3x^2/x^2} = \frac{4 + \frac{1}{x^2}}{\frac{2}{x^2} + 3}\]

Step 3: Apply the Limit to the Simplified Expression

Now apply the limit as \(x \to -\infty\). As \(x\) approaches \(-\infty\), both \(\frac{1}{x^2}\) and \(\frac{2}{x^2}\) approach 0. Thus, the expression simplifies to:\[\frac{4 + 0}{0 + 3} = \frac{4}{3}\]

Step 4: Conclude the Limit's Value

Now that the limit has been simplified, you can conclude that:\[\lim _{x \rightarrow -\infty} \frac{4 x^{2}+1}{2+3 x^{2}} = \frac{4}{3}\] This is the final answer for the limit of the given function as \(x\) approaches \(-\infty\).

Key Concepts

Precalculus

Precalculus serves as the bridge between algebra and calculus. It's an essential subject that prepares students for the concepts and skills required in calculus. In precalculus, we often deal with functions, which can include various types such as linear, polynomial, and rational functions. Understanding limits and how functions behave as they approach a particular point—in this case, infinity—is a fundamental aspect of precalculus.

Here, you'll use algebraic techniques to manipulate equations and expressions. These techniques include factoring, simplifying, and identifying key characteristics of functions. Such skills are crucial not only for handling limits but also for understanding concepts like continuity and rates of change you will encounter later in calculus. So, having a solid grasp of these precalculus skills will make the transition to more advanced topics much smoother.

Limits at infinity

Limits at infinity refer to the behavior of a function as the input value approaches positive or negative infinity. In mathematical terms, when we say \( \lim_{x \to -\infty} f(x) \), we are exploring what value \( f(x) \) trends towards as \( x \) becomes very large in the negative direction.

In our problem, the limit at infinity helps us observe how a rational function behaves as \( x \) dramatically decreases. This kind of limit is particularly insightful when discussing end behavior and asymptotes of graphs. Essentially, by examining limits at infinity, we can determine the horizontal asymptotes of the function, which are lines that the graph approaches but never touches as \( x \) moves towards infinity.

• To resolve such limits, focus on the terms with the highest powers of \( x \) in the numerator and the denominator since they dominate the other terms in the ratio.
• Simple division by the highest power of \( x \) in the denominator simplifies the function, making it easier to see the limit value.

Rational functions

Rational functions are ratios of polynomials, expressed in the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). These functions can have a wide range of behaviors and complexities, providing interesting usage in mathematical modeling and analysis.

One key aspect of rational functions is identifying and analyzing their horizontal asymptotes, which can be deduced from limits at infinity. The general rule of thumb involves comparing the degree—the highest power term—of the polynomials in the numerator and denominator:

• If the degree of \( P(x) \) is less than the degree of \( Q(x) \), the horizontal asymptote is \( y = 0 \).
• If the degrees are the same, the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients.
• If the degree of \( P(x) \) is greater than the degree of \( Q(x) \), there is no horizontal asymptote.

Understanding these rules not only aids in evaluating functions but also plays a pivotal role in graphing these kinds of functions effectively.