Problem 73 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In Exercises 73-80, find the indicated limits, if they exist. \(\lim _{x \rightarrow \infty} \frac{3 x+2}{x-5}\)

Step-by-Step Solution

Answer

The short answer to the given question \(\lim_{x \rightarrow \infty} \frac{3x + 2}{x - 5}\) is 3.

Step 1: Identify the function type

We are given a rational function with linear terms in the numerator and denominator: \(\lim_{x \rightarrow \infty} \frac{3x + 2}{x - 5}\).

Step 2: Simplify the function

Divide both numerator and denominator by the leading term of the denominator (x): \(\lim_{x \rightarrow \infty} \frac{3 + \frac{2}{x}}{1 - \frac{5}{x}}\).

Step 3: Apply limit properties

Now, we can apply limit properties to evaluate the limit. As x approaches infinity, the terms \(\frac{2}{x}\) and \(\frac{5}{x}\) will approach 0: \(\lim_{x \rightarrow \infty} \frac{3 + \frac{2}{x}}{1 - \frac{5}{x}} = \frac{3 + \lim_{x \rightarrow \infty} \frac{2}{x}}{1 - \lim_{x \rightarrow \infty} \frac{5}{x}} = \frac{3 + 0}{1 - 0}\).

Step 4: Calculate the final limit

Now, we have a simple expression: \(\frac{3 + 0}{1 - 0} = \frac{3}{1}\). Thus, the limit of the given function as x approaches infinity is: \(\lim_{x \rightarrow \infty} \frac{3x + 2}{x - 5} = 3\).

Key Concepts

Rational Functions

Rational functions are mathematical expressions formed by the ratio of two polynomial functions, defined as \f\(R(x) = \frac{p(x)}{q(x)}\f\), where both \f\(p(x)\f\) and \f\(q(x)\f\) are polynomials and \f\(q(x) eq 0\f\). In our exercise example, the function \f\(\frac{3x+2}{x-5}\f\) is a rational function, with a linear polynomial in the numerator and a linear polynomial in the denominator.

To understand the behavior of such functions, particularly as \f\(x\f\) approaches large values (\f\(x \rightarrow \f\) infinity), one must be familiar with their potential for undefined points (where \f\(q(x)=0\f\)) and asymptotic behavior (tendencies toward certain values without ever reaching them). Specifically, in the case of \f\(\frac{3x+2}{x-5}\f\), as \f\(x\f\) grows larger, the \f\(+2\f\) and \f\(-5\f\) terms become negligible compared to the terms that multiply \f\(x\f\), simplifying our understanding of the function's end behavior.

Limit Properties

The rules, or properties, of limits in calculus, are essential for evaluating how functions behave as they approach specific points or infinity. These properties are mathematical tools that allow us to simplify complex expressions and make it possible to find the limit without having to calculate the value of the function closer and closer to the point in question.

One such property is that if \f\(f(x)\f\) and \f\(g(x)\f\) are functions with limits that exist at a certain point, and \f\(c\f\) is a constant, then \f\(\f\)lim_{x \rightarrow c} [f(x) \fpm g(x)] = \f\(lim_{x \rightarrow c} f(x) \fpm \f\)lim_{x \rightarrow c} g(x)\f\( ' and '\f\)\f\(lim_{x \rightarrow c} cf(x) = c \f\)lim_{x \rightarrow c} f(x)\f\('. Additionally, provided that \f\)g(x) eq 0\f\(, \f\)\f\(lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \frac{\f\)lim_{x \rightarrow c} f(x)}{\f\(lim_{x \rightarrow c} g(x)}\f\)'. These properties were applied in our exercise to separate the constants from the variables that depended on x and to evaluate the limits of each piece independently.

Evaluating Limits

Evaluating limits is a fundamental aspect of calculus, wherein one determines the value that a function approaches as the input approaches a certain value. Notably, evaluating limits of rational functions involves certain strategies, such as factorization, simplification, and the use of limit properties.

In the given exercise, once we simplified the rational function by dividing by \f\(x\f\), we could observe that as \f\(x\f\) approaches infinity, the terms \f\(\frac{2}{x}\f\) and \f\(\frac{5}{x}\f\) approach zero, effectively leaving us with the simple ratio of \f\(\frac{3}{1}\f\). This approach demystifies the process behind evaluating limits and showcases how limit properties can streamline the calculation of a limit for more complex functions. Furthermore, understanding these concepts will allow students not only to solve textbook problems but also to analyze and interpret the behavior of functions in a broader mathematical and real-world context.