Problem 48 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit. $$\lim _{x \rightarrow \infty} \frac{2 x^{3}-x-3}{6 x^{2}-x-1}$$

Step-by-Step Solution

Answer

The limit is \( \frac{1}{3} \).

Step 1: Identify the highest degree term in the numerator and the denominator

The given function is \( \frac{2x^3 - x - 3}{6x^2 - x - 1} \). Notice that the highest degree term in the numerator is \( 2x^3 \) and in the denominator is \( 6x^2 \).

Step 2: Factor out the highest degree term from the numerator and the denominator

Divide each term in the numerator by \( x^3 \) and each term in the denominator by \( x^2 \). This gives \( \frac{2 - \frac{1}{x^2} - \frac{3}{x^3}}{6 - \frac{1}{x} - \frac{1}{x^2}} \).

Step 3: Simplify the expression as \( x \rightarrow \infty \)

As \( x \rightarrow \infty \), the terms \( \frac{1}{x} \), \( \frac{1}{x^2} \), and \( \frac{3}{x^3} \) approach 0. Therefore, the expression simplifies to \( \frac{2}{6} \).

Step 4: Compute the limit

After simplifying, the limit \( \frac{2}{6} \) simplifies to \( \frac{1}{3} \). Therefore, \( \lim_{x \to \infty} \frac{2x^3 - x - 3}{6x^2 - x - 1} = \frac{1}{3} \).

Key Concepts

Limits

In calculus, limits are a fundamental concept used to understand the behavior of functions as their input approaches a certain value or even infinity. They help us grasp how a function behaves near a specific point, which is crucial for defining derivatives and integrals.

• When we say \( \lim_{x \to \infty} f(x) \), we are interested in what happens to \( f(x) \) as \( x \) becomes very large.
• If a limit exists, it represents a value that \( f(x) \) gets closer and closer to, without necessarily reaching it.

In the limit expression \( \lim_{x \to \infty} \frac{2x^3 - x - 3}{6x^2 - x - 1} \), we analyzed the behavior of a rational function with the variable \( x \) heading towards infinity. By examining larger and larger values of \( x \), we utilized algebraic simplification techniques to determine that this limit equals \( \frac{1}{3} \). This gives us significant insight into the function's end behavior as \( x \) increases.

Rational functions

Rational functions are expressions that are the ratio of two polynomials. They can exhibit a wide variety of behaviors, making them important to understand in calculus.

• A general rational function is written as \( \frac{N(x)}{D(x)} \), where \( N(x) \) and \( D(x) \) are polynomials.
• The degree of a polynomial is the highest power of \( x \) in the expression.

In our example, \( \frac{2x^3 - x - 3}{6x^2 - x - 1} \), the numerator is a cubic polynomial (degree 3), while the denominator is a quadratic polynomial (degree 2). Knowing the degrees of these polynomials allows us to simplify and find the limit. As \( x \to \infty \), we focus on these highest degree terms, largely because they dominate the growth rate of the rest of the polynomial's terms.

End behavior

End behavior in calculus describes how a function behaves as its input becomes very large (positively or negatively). This often involves determining the leading factors of the function.

• The end behavior can tell us if a function approaches a horizontal asymptote, diverges, or converges to a finite limit.
• For rational functions, the end behavior is most influenced by the comparison of the polynomial degrees in the numerator and denominator.

In examining the function's end behavior \( \frac{2x^3 - x - 3}{6x^2 - x - 1} \), we recognized that the numerator's degree being higher than the denominator's degree implies the function grows without bound as \( x \to \infty \).However, by factoring out the highest powers and simplifying, it revealed an end behavior approaching a horizontal line, specifically \( y = \frac{1}{3} \). This limit tells us that no matter how large \( x \) gets, the function's value stabilizes to \( \frac{1}{3} \).