Problem 46 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Degrees of the Polynomials

Look at the degrees of the polynomials in the numerator and the denominator. - The degree of the numerator \(3x^3 + 2x - 1\) is 3.- The degree of the denominator \(2x^4 - 3x^3 - 2\) is 4.

Step 2: Evaluate the Behavior of the Function

As \(x\) approaches infinity, consider the leading terms since they dominate the growth of the polynomials.- The leading term in the numerator is \(3x^3\).- The leading term in the denominator is \(2x^4\).

Step 3: Simplify the Expression

Since the degrees of the numerator and the denominator are different, divide both the numerator and denominator by \(x^4\), the highest power of \(x\) in the expression:\[\frac{3x^3}{x^4} + \frac{2x}{x^4} - \frac{1}{x^4} = \frac{3}{x} + \frac{2}{x^3} - \frac{1}{x^4}\]\[\frac{2x^4}{x^4} - \frac{3x^3}{x^4} - \frac{2}{x^4} = 2 - \frac{3}{x} - \frac{2}{x^4}\]

Step 4: Take the Limit

As \(x\) approaches infinity, each term that has \(x\) in the denominator goes to zero:\[\lim_{x \to \infty} \left( \frac{3}{x} + \frac{2}{x^3} - \frac{1}{x^4} \right) = 0\]\[\lim_{x \to \infty} \left( 2 - \frac{3}{x} - \frac{2}{x^4} \right) = 2\]

Step 5: Calculate the Final Limit

Divide the results from the previous step:\[\frac{0}{2} = 0\]Therefore, the limit of the expression as \(x\) approaches infinity is 0.

Key Concepts

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to whole number powers and are multiplied by coefficients. They form the backbone of many algebraic and calculus problems. A typical polynomial looks like this:

• The term leading coefficient represents the number multiplying the highest power of the variable.
• The degree of a polynomial is the highest exponent found among the terms.

Consider the polynomial in our numerator, which is \(3x^3 + 2x - 1\). This is a polynomial of degree 3 because the highest power of \(x\) is 3. Similarly, the polynomial in the denominator \(2x^4 - 3x^3 - 2\) has a degree of 4 since the highest power is \(x^4\).
This concept of degree is essential because as variables grow very large or very small, the term with the highest power will dominate the behavior of the function. Understanding this helps in simplifying calculations like limits.

Rational Functions

A rational function is essentially a fraction where both the numerator and the denominator are polynomials. The study of these functions is particularly interesting as their behavior can change dramatically with large or small values of \(x\).
In our example, we have a rational function represented as:

• The numerator is \(3x^3 + 2x - 1\), a polynomial of degree 3.
• The denominator is \(2x^4 - 3x^3 - 2\), a polynomial of degree 4.

When considering limits, especially as \(x\) approaches infinity, the key is to focus on the highest degree terms. This helps in determining how the function behaves or "simplifies" at extreme values of \(x\). By dividing every term by \(x^4\), which is the highest degree in the denominator, we observed that most terms approach zero, leading to a simpler expression to evaluate.

Limit at Infinity

Limits help us understand the behavior of a function as the input approaches a particular value. When we talk about limits at infinity, we want to know what happens as \(x\) gets larger and larger—approaching infinity.
In the given problem, evaluating the limit at infinity for the rational function \(\frac{3x^3 + 2x - 1}{2x^4 - 3x^3 - 2}\) involved simplifying the expression to see dominance. By taking the limit of each term:

• Terms like \(\frac{3}{x}\), \(\frac{2}{x^3}\), and \(\frac{1}{x^4}\) shrink toward zero because the denominator gets infinitely large while the numerators stay constant.
• Similarly, \(2 - \frac{3}{x} - \frac{2}{x^4}\) simplifies to 2 as \(x\) tends to infinity.

Ultimately, when we simplify and calculate the limit, it culminates in evaluating basic fractions like \(\frac{0}{2}\), which results in zero. This concept is useful in calculus for predicting long-term behavior of functions.