Problem 25 - Calculus: Early Transcendentals

Question

Determine the following limits. $$\lim _{x \rightarrow \infty} \frac{14 x^{3}+3 x^{2}-2 x}{21 x^{3}+x^{2}+2 x+1}$$

Step-by-Step Solution

Answer

Answer: The limit of the given rational function as x approaches infinity is $\frac{2}{3}$.

Step 1: Identify the highest power of x in the rational function

The highest power of x in both the numerator and denominator is x^3.

Step 2: Divide each term in the function by the highest power of x

We'll divide each term in the numerator and denominator by x^3: $$\frac{14x^3 + 3x^2 - 2x}{21x^3 + x^2 + 2x + 1} \cdot \frac{\frac{1}{x^3}}{\frac{1}{x^3}} = \frac{14 + \frac{3}{x} - \frac{2}{x^2}}{21 + \frac{1}{x} + \frac{2}{x^2} + \frac{1}{x^3}}$$

Step 3: Take the limit as x approaches infinity

As x approaches infinity, the terms with x in the denominator will approach zero: $$\lim_{x\to\infty} \frac{14 + \frac{3}{x} - \frac{2}{x^2}}{21 + \frac{1}{x} + \frac{2}{x^2} + \frac{1}{x^3}} = \frac{14 + 0 - 0}{21 + 0 + 0 + 0} = \frac{14}{21}$$

Step 4: Simplify the answer

Finally, we'll simplify the fraction: $$\frac{14}{21} = \frac{2\cdot7}{3\cdot7} = \frac{2}{3}$$ So the limit of the given rational function as x approaches infinity is: $$\lim_{x\to\infty} \frac{14x^3 + 3x^2 - 2x}{21x^3 + x^2 + 2x + 1} = \boxed{\frac{2}{3}}$$

Key Concepts

Rational Functions

When we deal with rational functions, we're essentially looking at ratios of polynomials. In simpler terms, a rational function is a fraction where both the numerator (the top part) and the denominator (the bottom part) are polynomials. Take for example the function \( \frac{14x^3 + 3x^2 - 2x}{21x^3 + x^2 + 2x + 1} \), which is the subject of our limit problem. It has a polynomial of degree three, or a cubic polynomial, in both the numerator and denominator.

Understanding the behavior of polynomials can give us insight into the behavior of the rational function as a whole, especially as we evaluate limits. As the values of \( x \) become very large (approaching infinity), the highest power of \( x \) typically dominates the growth of the polynomial, essentially determining its long-term behavior.

Limit to Infinity

The limit to infinity refers to the value that a function approaches as the variable within it goes toward positive or negative infinity. In other words, it’s like asking, 'As I go farther and farther along on the x-axis, what value is my function getting close to?'

In our example, we looked at \( \lim_{x \to \infty} \frac{14x^3 + 3x^2 - 2x}{21x^3 + x^2 + 2x + 1} \). As \( x \) grows larger and larger, terms with \( x \) in the denominator become negligible. That's why, after some simplification, terms like \( \frac{3}{x} \) and \( \frac{2}{x^2} \) drop out because they approach zero. What we're left with is the ratio of the coefficients of the highest power of \( x \) from the numerator and denominator to determine the limit at infinity.

Simplifying Expressions

Simplifying an expression means rewriting it in an easier or more comprehensible form without changing its value. It usually involves cancelling out factors, combining like terms, and reducing fractions to their lowest terms. In our limit problem, we simplified the fraction \( \frac{14}{21} \) by identifying the common factor, which was 7, and then dividing both the numerator and the denominator by this number to get \( \frac{2}{3} \) as the simplest form of the fraction.

Simplification is a valuable skill in calculus, as it makes it easier to understand and interpret the results of limit problems, integrals, derivatives, and more. When simplifying, always look for common factors, and remember that terms with \( x \) in the denominator become insignificant as \( x \) approaches infinity, which can greatly streamline the problem-solving process.