Question

\(\operatorname{Lt}_{x \rightarrow \infty} \frac{4 x-3}{(2 x+3)}=\) (1) 0 (2) 1 (3) \(\frac{1}{2}\) (4) 2

Short answer

Answer: The limit of the function as x approaches infinity is 2.

Step-by-step solution

Identify the highest degree terms

In our function \(\frac{4x - 3}{2x + 3}\), the highest degree terms are \(4x\) in the numerator and \(2x\) in the denominator.

Divide the numerator and denominator by the highest degree term of the denominator

Also known as the horizontal asymptote method, divide the numerator and denominator by the highest degree term of the denominator, which is 2x, to simplify the expression: \( \frac{\frac{4x}{2x} - \frac{3}{2x}}{1 + \frac{3}{2x}} \)

Simplify the expression

After dividing, simplify the expression: \( \frac{2 - \frac{3}{2x}}{1 + \frac{3}{2x}} \)

Determine the limit as x approaches infinity

As x approaches infinity, terms with x in the denominator will approach 0. So, both \(\frac{3}{2x}\) in the numerator and \(\frac{3}{2x}\) in the denominator approach 0: \( \operatorname{Lt}_{x \rightarrow \infty} \frac{2 - \frac{3}{2x}}{1 + \frac{3}{2x}} \rightarrow \frac{2 - 0}{1 + 0} = \frac{2}{1} \)

Compare the result to the given options

Our limit as x approaches infinity is \(\frac{2}{1}\), which simplifies to 2. Comparing our result to the given options, we see that the correct answer is option (4) 2.

Key concepts

Asymptotes

When studying the behavior of functions as the input values grow larger and larger—or smaller and smaller—we often encounter the concept of asymptotes. An asymptote is a line that a graph of a function approaches as the input or output grows without bound. Think of asymptotes as the 'horizon' for the journey of the graph; it's where the graph is headed, but typically never quite reaches.

There are three types of asymptotes: vertical, horizontal, and oblique (sometimes referred to as slant). A vertical asymptote occurs when the function heads towards infinity or negative infinity as the input approaches a certain value. A horizontal asymptote reflects the behavior of a graph as the input goes to infinity or negative infinity, showing us the output level that the function is approaching. Finally, an oblique asymptote is a straight line that the graph approaches that is neither parallel to the x-axis nor to the y-axis.

Understanding the concepts of asymptotes is crucial when analyzing the limits and long-term behavior of functions, and is particularly handy when dealing with rational functions, as is the case in the exercise provided.

Horizontal Asymptote Method

The horizontal asymptote method is a technique used to determine the end behavior of a function—specifically, to find horizontal asymptotes of rational functions. In simple terms, it's a way to gauge where the graph of the function will 'settle down' as the input gets arbitrarily large or small.

To utilize this method, we should start by comparing the degrees of the polynomials in the numerator and denominator. If the degrees are the same, the horizontal asymptote will be the ratio of the coefficients of the highest degree terms. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the x-axis (y=0). And if the numerator's degree is greater, there is no horizontal asymptote (there may be an oblique asymptote instead).

Applying the Horizontal Asymptote Method

In the exercise, by identifying and dividing the highest degree terms of the numerator and the denominator, the problem was simplified. This step was pivotal for finding the limit at infinity. When the terms involving x in the denominator were simplified out, the resulting constant term revealed the location of our horizontal asymptote, allowing us to solve for the limit as x approaches infinity.

Simplifying Expressions

The process of simplifying expressions is vital in calculus, especially when trying to find limits. Simplifying can involve a variety of techniques such as factoring, canceling, expanding, or—most common in finding limits—dividing out terms. The main goal of simplification is to make a complex expression more manageable and easier to analyze.

In the context of finding limits at infinity, like in our exercise, simplifying expressions often involves dividing terms out to remove the variable (often x) from the denominator, which allows us to more accurately see the behavior of the function as x becomes very large.

Why Simplify?

By simplifying, we can remove components of the expression that become insignificant as x approaches infinity, such as fractions where x is in the denominator. This was demonstrated in the given solution, where terms like \(\frac{3}{2x}\) approached zero and effectively had no influence on the limit of the function as x approached infinity. Simplifying allowed us to reduce the function to a form where the limit could be clearly seen and easily computed.