Problem 64 - Precalculus

Question

For Exercises 61 through \(64,\) evaluate the limits by dividing the numerator and denominator by the highest power of \(x\) occurring in the denominator. $$\lim _{x \rightarrow-\infty} \frac{8 x^{3}-27}{x^{4}-1}$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Highest Power in the Denominator

In the given expression \(\frac{8x^3 - 27}{x^4 - 1}\), the highest power of \(x\) in the denominator is \(x^4\). We need to divide both the numerator and denominator by \(x^4\) to simplify the limit as \(x\) approaches \(-\infty\).

Step 2: Rewrite the Expression

Divide both the numerator and the denominator by the highest power in the denominator (\(x^4\)): \[\frac{8x^3 - 27}{x^4 - 1} = \frac{\frac{8x^3}{x^4} - \frac{27}{x^4}}{\frac{x^4}{x^4} - \frac{1}{x^4}} = \frac{\frac{8}{x} - \frac{27}{x^4}}{1 - \frac{1}{x^4}}.\]

Step 3: Evaluate each Term as \(x \to -\infty\)

As \(x\) approaches \(-\infty\), evaluate each component of the fraction: - \(\frac{8}{x}\) approaches 0 because \(8\) divided by an infinitely large \(x\) is a very small value.- \(\frac{27}{x^4}\) approaches 0 because \(x^4\) grows much larger than 27.- \(\frac{1}{x^4}\) approaches 0 as well for the same reason.

Step 4: Calculate the Limit

By substituting the evaluated limits into the expression, we have: \[\lim _{x \to -\infty} \frac{\frac{8}{x} - \frac{27}{x^4}}{1 - \frac{1}{x^4}} = \frac{0 - 0}{1 - 0} = 0.\]Therefore, the limit is 0.

Key Concepts

Rational Functions

Rational functions are mathematical expressions represented as the ratio of two polynomials. They take the general form \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials. These types of functions can exhibit interesting behavior as variables approach specific values or even infinity. The degree of the polynomials, or the highest power of \(x\) in each, greatly influences the characteristics of the rational function.
When studying rational functions, focuses include:

• The degree of the numerator and denominator.
• Finding points of undefined behavior (where the denominator equals zero).
• Create graphs that help visualize behavior at specific points, like where they may approach infinity.

Identifying the highest power of \(x\) is especially critical because it determines the leading term, which in turn influences the long-term behavior of the function as \(x\) gets very large or very small. Analyzing the limit of a rational function as \(x\) approaches infinity reveals long-term tendencies. This understanding is the first step in diving deeper into the limit evaluation.

Infinity Behavior

Understanding infinity behavior means finding how a function behaves as \(x\) approaches very large positive or negative values. For rational functions, like the one presented in the exercise, this usually involves analyzing the degrees of the polynomials in the numerator and denominator.
The steps involved:

• If the degree of the numerator is less than the degree of the denominator, the fraction will approach zero.
• If the degree of the numerator equals the degree of the denominator, the fraction approaches the ratio of the coefficients of the highest degree terms.
• If the degree of the numerator is greater than the degree of the denominator, the function approaches infinity or negative infinity.

For the current exercise, where \(\lim _{x \to -\infty} \frac{8x^3 - 27}{x^4 - 1}\) is evaluated, the degree of the numerator \((3)\) is less than the degree of the denominator \((4)\). Thus, the function approaches zero as \(x\) moves towards mothingus negative infinity.

Asymptotic Analysis

Asymptotic analysis in calculus involves looking at how functions behave as they tend toward infinity or a point of discontinuity. It gives us insights into the end behavior of a function. In the context of rational functions, we often seek to simplify the expression by dividing both the numerator and denominator by the highest power of \(x\) present in the denominator. This step simplifies the evaluation of the limit for \(x\) approaching infinity.
Here are the key ideas:

• Identify the leading term in the denominator. Divide both components of the expression by this term.
• Evaluate each term separately, considering their behavior as \(x\) approaches infinity or negative infinity.
• Common simplifying assumptions include terms approaching zero if they are fractions with \(x\) in the denominator and \(x\) nearing infinity.

Through these principles, we can take complex-looking rational functions and make them manageable, arriving at a result that reveals the function's asymptotic nature. This is critical in finding limits, predicting behavior patterns at infinity, and understanding how functions interact in broader contexts.