Problem 55 - Calculus: Early Transcendentals

Question

Let \( P \) and \( Q \) be polynomials. Find $$ \lim_{x \to \infty} \frac{P(x)}{Q(x)} $$ if the degree of \( P \) is (a) less than the degree of \( Q \) and (b) greater than the degree of \( Q \).

Step-by-Step Solution

Answer

(a) 0; (b) Infinity.

Step 1: Identify the degrees of the polynomials

Let the degree of \( P(x) \) be \( m \) and the degree of \( Q(x) \) be \( n \). We need to analyze cases separately: \( m < n \) and \( m > n \).

Step 2: Case (a): Degree of P is less than the degree of Q

If \( m < n \), as \( x \to \infty \), the terms with the highest degree in \( Q(x) \) will dominate both the numerator and the denominator. The leading term of \( P(x) \) will have a lower degree and thus will approach zero compared to the highest degree term in \( Q(x) \). Therefore, \[ \lim_{x \to \infty} \frac{P(x)}{Q(x)} = 0. \]

Step 3: Case (b): Degree of P is greater than the degree of Q

If \( m > n \), as \( x \to \infty \), the leading term of \( P(x) \) will dominate the expression since it has a higher degree than the leading term of \( Q(x) \). The polynomial \( \frac{P(x)}{Q(x)} \) behaves like a polynomial of degree \( m-n \), thus the limit approaches infinity. So, \[ \lim_{x \to \infty} \frac{P(x)}{Q(x)} = \infty. \]

Key Concepts

Degree of Polynomials

The degree of a polynomial is the highest power of the variable in the expression. For example, in the polynomial \( 3x^4 + 2x^3 + x + 7 \), the degree is 4 because the term with the highest power is \( x^4 \). The coefficient of the term with the highest degree is significant since it often determines the polynomial's behavior for very large or very small values of \( x \). The degree helps us understand the polynomial's growth rate. As \( x \) increases or decreases, terms with lower degrees become negligible compared to the highest degree term. This property is crucial when evaluating limits like \( \lim_{x \to \infty} \frac{P(x)}{Q(x)} \). Knowing which polynomial has a higher degree allows us to easily determine whether the fraction approaches zero, infinity, or some other value.

Polynomial Division

Polynomial division is a process similar to long division in arithmetic. It involves dividing a polynomial by another polynomial to simplify the expression, often resulting in a quotient and a remainder. This technique is vital for understanding the limits of ratios of polynomials.When comparing two polynomials \( P(x) \) and \( Q(x) \), if both polynomials have a leading term, the division helps identify which term dominates as \( x \to \infty \). For instance, if the polynomial \( P(x) \) is divided by \( Q(x) \), the comparison of their degrees dictates the outcome. If \( P(x) \) is of a higher degree, the quotient represents a polynomial of that excess degree. In contrast, if \( Q(x) \) is of a higher degree, the division results in a very small number approaching zero.Understanding polynomial division provides clarity on the interplay between numerator and denominator, especially when their degrees dictate the limit behavior.

Asymptotic Behavior

Asymptotic behavior examines how functions behave as the input approaches a particular value, often infinity. For polynomials when \( x \to \infty \),

• If the degree of the polynomial in the numerator is less than the degree in the denominator, the function approaches zero.
• If the numerator's degree is greater, the function approaches infinity because the numerator grows faster than the denominator.

This behavior is governed by the terms with the highest degrees. They are the dominant factors because they grow at a rate much faster than other terms as \( x \to \infty \). This concept helps us predict and understand the end behavior of rational functions, leading to practical uses in real-world applications like physics and engineering where such limits often represent asymptotic constraints or growth limits.