Problem 5 - Calculus for Biology and Medicine

Question

Evaluate the limits in problems. $$ \lim _{x \rightarrow \infty} \frac{1-x^{3}+2 x^{4}}{2 x^{2}+x^{4}} $$

Step-by-Step Solution

Answer

The limit is 2.

Step 1: Analyze the Degree of the Polynomial

To evaluate the limit as \( x \) approaches infinity, first compare the degrees of the polynomials in the numerator and the denominator. The numerator is \( 1-x^{3}+2x^{4} \) with a highest degree term of \( x^4 \), and the denominator is \( 2x^{2}+x^{4} \) with a highest degree term also of \( x^4 \). Since both have the same highest degree, the limit will be determined by the coefficients of these highest degree terms.

Step 2: Extract Leading Coefficients

Identify and extract the leading coefficients of the highest degree terms from both the numerator and denominator. The coefficient of \( x^4 \) in the numerator is 2, and the coefficient of \( x^4 \) in the denominator is 1.

Step 3: Simplify the Expression

Since only the terms with the highest degree will significantly affect the limit as \( x \rightarrow \infty \), simplify the expression by considering only these terms: \( \frac{2x^{4}}{x^{4}} = \frac{2}{1} \).

Step 4: Evaluate the Limit

Evaluate the simplified expression. The limit evaluates to the fraction of the leading coefficients: \( \lim_{x \rightarrow \infty} \frac{2}{1} = 2 \).

Key Concepts

Understanding Polynomials

Polynomials are mathematical expressions involving variables, coefficients, and non-negative integer exponents. They look like this: \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where \( a_n, a_{n-1}, \ldots, a_0 \) are constants, often called coefficients, and \( x \) represents a variable.

A key characteristic of polynomials is that they can vary greatly based on their degree, which is simply the highest exponent of the variable. Polynomials are smooth and continuous, meaning they can be drawn as one unbroken curve. Some examples include quadratic polynomials like \( x^2+2x+1 \), cubic polynomials such as \( x^3 - 4x \), and quartic polynomials like \( x^4 - 3x^3 + x + 9 \).

When working with polynomials in calculus, particularly when dealing with limits, it is essential to compare their degrees to understand how they behave as the variable \( x \) approaches infinity. In these cases, higher degree terms will dictate the polynomial's growth.

Explaining Leading Coefficients

In any polynomial, the leading coefficient is the number multiplying the term with the highest degree of the variable. For instance, in \( 2x^4 - 3x^3 + x - 6 \), the leading coefficient is 2, since it multiplies the term \( x^4 \), which is the term with the highest degree.

Leading coefficients are crucial in evaluating limits, particularly when determining the behavior of a function as \( x \) approaches infinity. When comparing two polynomials of the same degree, the behavior of their ratio, as \( x \) becomes very large, is primarily determined by the ratio of their leading coefficients.

For example, if we take the expression \( \lim_{x \to \infty} \frac{ax^n}{bx^n} \), since both the numerator and the denominator have the same degree, the limit simplifies to \( \frac{a}{b} \). This highlights the significance of leading coefficients in calculus problems involving limits.

Degrees of Polynomials and Their Importance

The degree of a polynomial is an essential concept in understanding how the function behaves, particularly as the variable approaches extreme values like infinity. The degree is simply the highest power of the variable present in the polynomial. For example, \( 5x^3 + 2x^2 + 7 \) is a third-degree polynomial because the highest power of \( x \) is 3.

In the context of limits, the degree of polynomials is pivotal. When evaluating limits where the input approaches infinity, compare the degree of the polynomials in the numerator and the denominator. If the degrees are the same, the limit is determined by the leading coefficients, which determine the function's direction at large scales.

This means if the degree of the polynomial in the numerator is greater than the degree in the denominator, the function will tend towards infinity or negative infinity. Conversely, if the degree in the denominator is larger, the function's limit is zero. Understanding this makes it easier to tackle limit problems involving polynomials.