Problem 22 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(19-38,\) find the limit. $$ \lim _{x \rightarrow-\infty} \frac{4 x^{2}+5}{x^{2}+3} $$

Step-by-Step Solution

Answer

The limit as \(x\) approaches negative infinity of the given function is 4.

Step 1: Identify the degrees of the numerator and denominator

The degree of the numerator and denominator are both 2 since the highest power of \(x\) is 2 in both the numerator (from \(4x^2\)) and the denominator (from \(x^2\)).

Step 2: Apply the rule for limits at infinity for rational functions

Given that the degree of the numerator and denominator are the same, the limit is the ratio of the leading coefficients of the numerator and the denominator. The leading coefficient of the numerator is 4 and the leading coefficient of the denominator is 1.

Step 3: Compute the Limit

Using the rule from Step 2, the limit is then the coefficient of \(x^2\) in the numerator divided by the coefficient of \(x^2\) in the denominator, therefore, \(\lim_{x \rightarrow -\infty} \frac{4x^2 + 5}{x^2 + 3} = \frac{4}{1} = 4\).

Key Concepts

Limits at Infinity

Understanding the concept of limits at infinity is crucial when dealing with the behavior of functions as they approach very large positive or negative values. It helps us to describe the end behavior of a function, essentially telling us what value the function is getting closer to as the input grows infinitely large or decreases infinitely large in the negative sense.

In the context of our exercise, we evaluate the limit of a rational function as the variable approaches negative infinity. This exploration informs us about how the function behaves as the values of the variable become large in a negative direction. When we say \( \lim_{x \rightarrow -\infty} \) for a function, we're interested in the value that the function approaches, not the value it actually reaches, as infinity is a concept, not a number.

Rational Functions

Rational functions are algebraic expressions that represent the division of two polynomials. For instance, the function given in the exercise \( \frac{4x^2 + 5}{x^2 + 3} \) is a rational function because it can be expressed as a ratio of two polynomials where the numerator is \( 4x^2 + 5 \) and the denominator is \( x^2 + 3 \).

When finding limits of rational functions as \( x \) approaches infinity (positive or negative), we often use the behavior of the leading terms to determine the end behavior of the function. Moreover, if the highest power of \( x \) is the same in both the numerator and the denominator—as in the case of our exercise—the limit will simply be the ratio of the coefficients of those terms, assuming the coefficients are nonzero.

Leading Coefficients

The leading coefficient of a polynomial is the coefficient of the term with the highest power of the variable. In our exercise, we see \( 4x^2 \) and \( x^2 \) as the leading terms of the numerator and the denominator, respectively. The coefficients for these terms are 4 and 1.

When evaluating limits at infinity for rational functions where the highest power of \( x \) is the same in the numerator and the denominator, the limit is the ratio of the leading coefficients. This shortcut is incredibly useful because it allows us to bypass more complex calculations, providing a quick and elegant way to find the limit. For the given exercise, simplifying the behavior of the function to just its leading coefficients reveals that the limit as \( x \rightarrow -\infty \) is 4, which is the ratio of the leading coefficient of the numerator to that of the denominator.