Question

Find the limit. \(\lim _{x \rightarrow \infty} \frac{4 x-3}{2 x+1}\)

Short answer

The limit of the function as x approaches infinity is 2

Step-by-step solution

Identify the degree of the polynomials

In the given function \(\frac{4x-3}{2x+1}\), the degree of the polynomial in the numerator is 1 (from the term \(4x\)) and that of the denominator is also 1 (from the term \(2x\)). Hence, these two polynomials are of equal degree.

Identify the leading coefficients

The leading coefficient is the coefficient of the term with highest degree in the polynomial. In this case, in the numerator, the leading coefficient is 4 and in the denominator, the leading coefficient is 2.

Compute the limit

Based on the rule mentioned in the analysis, for rational functions where the numerator and the denominator have the same degree, the limit as x approaches infinity is simply the ratio of the leading coefficients. So, \(\lim_{x \rightarrow \infty} \frac{4x-3}{2x+1} = \frac{4}{2} = 2\)

Key concepts

Rational Functions

Rational functions are a specific type of function that can be expressed as the ratio of two polynomials. They are written in the form \( \frac{P(x)}{Q(x)} \), where each of \( P(x) \) and \( Q(x) \) are polynomials with \( Q(x) eq 0 \). The degree of a polynomial is determined by the highest power of \( x \) present in the polynomial.
One important aspect of rational functions is the behavior of the function as \( x \) approaches certain values, such as infinity. Analyzing the limits of rational functions helps in understanding their asymptotic behavior, which describes how the function behaves as it gets very large or very small.
In the given exercise, we explore a rational function with a polynomial of degree 1 both in the numerator and the denominator, which is a simple but fundamental form of rational functions. They often serve as a stepping stone to understanding more complex rational functions and their behaviors at infinity.

Leading Coefficients

In any polynomial, the leading coefficient is the coefficient of the term with the highest degree. For instance, in the polynomial \( ax^n + bx^{n-1} + \ldots + cx + d \), the leading coefficient would be \( a \), assuming \( a eq 0 \). Leading coefficients are crucial when analyzing the limits of rational functions as \( x \) approaches infinity.

Determining Importance

When the numerator and denominator of a rational function have the same degree, as \( x \) tends to infinity, the smaller degree terms become insignificant. In such cases, the limit depends primarily on the leading coefficients. This simplification provides a straightforward way to calculate limits for these specific types of rational functions.
For example, in the expression \( \frac{4x - 3}{2x + 1} \), the leading terms are \( 4x \) in the numerator and \( 2x \) in the denominator, meaning the limit as \( x \) approaches infinity will be the ratio of the leading coefficients: \( \frac{4}{2} = 2 \).

Infinity

Infinity is a concept in mathematics that describes something without any limit or end. It's often used in limits to describe how a function behaves as it grows larger and larger. In analysis, we use infinity to understand how functions behave as their input values increase without bound.

Handling Infinity in Limits

When dealing with rational functions, understanding how they behave as \( x \) approaches infinity is often crucial. The key to working with infinity in these contexts is recognizing that terms with lower degrees become negligible compared to terms with higher degrees. This is why, in the context of rational functions, the degree and leading coefficients become particularly important.
While mathematically treating infinity requires careful handling, conceptually, thinking of it as a boundary helps to understand these phenomena. It's crucial in determining the asymptotic behavior of functions and predicting their long-term behavior.