Problem 15 - Calculus Early Transcendentals

Question

\(15-36\) Find the limit. $$ \lim _{x \rightarrow \infty} \frac{1}{2 x+3} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Expression

We're given the limit to solve: \( \lim_{x \rightarrow \infty} \frac{1}{2x + 3} \). This asks us to find the value this expression approaches as \( x \) becomes infinitely large.

Step 2: Simplify the Expression

As \( x \) approaches infinity, the term \( 2x \) in the denominator becomes significantly larger than the constant \( 3 \). So, the denominator \( 2x + 3 \) behaves approximately like \( 2x \).

Step 3: Substitute Large Values of x

As \( x \) becomes very large, say approaching infinity, the fraction becomes \( \frac{1}{2x} \). In this expression, the \( 2x \) in the denominator grows infinitely large.

Step 4: Evaluate the Limit

With the simplified expression \( \frac{1}{2x} \), as \( x \) becomes infinitely large, the fraction approaches zero because a finite number divided by an infinitely large number results in a value that approaches zero.

Key Concepts

Understanding Infinite Limits

Infinite limits are a fundamental concept in calculus that deal with what happens to a function as its input, typically denoted by \( x \), goes to infinity or negative infinity. Specifically, an infinite limit helps us understand the end behavior of a function. In other words, it tells us the value a function is approaching as the input grows without bound.
For instance, with the expression \( \lim_{x \rightarrow \infty} \frac{1}{2x + 3} \), we are interested in knowing how the value of \( \frac{1}{2x + 3} \) behaves as \( x \) keeps getting larger and larger.

This concept is essential because it allows us to predict the behavior of functions at the far ends of their domain. By evaluating infinite limits, we gain insights into whether a function heads toward a specific value, infinite values, or even shows no trend.

Mastering Limit Evaluation

Evaluating limits is a crucial skill in calculus that involves finding the limit of a function as its input approaches some value, such as zero, infinity, or any real number. The process involves analyzing the behavior of the function near that specific point.

In our case with \( \lim_{x \rightarrow \infty} \frac{1}{2x + 3} \), we take specific steps:

• Recognize the dominant term in the denominator, which is \( 2x \), because, as \( x \) becomes extremely large, \( 3 \) becomes negligible compared to \( 2x \).
• Simplify the expression to focus on the substantial parts, here approximating it as \( \frac{1}{2x} \).
• Evaluate this simplified limit to see that as \( x \) increases, \( \frac{1}{2x} \) approaches zero.

Effective limit evaluation often involves simplification and using limit properties to find the value the function tends toward at its boundary.

Exploring Rational Functions

Rational functions are quotients of two polynomials and are represented as \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. Understanding and analyzing rational functions can reveal much about the behavior of various mathematical models.

When working with rational functions, especially as in our exercise \( \frac{1}{2x + 3} \), a common approach is to identify which part of the function dominates as \( x \) goes to infinity. In this instance, the linear term in the denominator, \( 2x \), eventually dwarfs any constants like \( 3 \).
As the degree of the denominator is higher than the numerator, the function's value decreases as x grows, leading the limit to become zero. This typical behavior of rational functions aids in predicting outcomes across a wide range of problems, making them a powerful tool in calculus and mathematical modeling.
Through studying rational functions, learners gain the ability to not only handle limits but also foresee the behavior of more complex systems and models.