Problem 55 - Calculus

Question

Assuming that limits as \(x \rightarrow \infty\) have the properties listed for limits as \(x \rightarrow c\) on page \(60,\) use algebraic manipulations to evaluate lim for the functions in Problems \(54-63\). $$f(x)=\frac{\pi+3 x}{\pi x-3}$$

Step-by-Step Solution

Answer

The limit is \( \frac{3}{\pi} \) as \( x \to \infty \).

Step 1: Identify the Highest Power of x

In the given function \( f(x) = \frac{\pi + 3x}{\pi x - 3} \), identify the highest power of \( x \) in both the numerator and the denominator. The highest power of \( x \) in both the numerator and the denominator is \( 1 \).

Step 2: Divide by the Highest Power of x

Divide every term in both the numerator and the denominator by \( x \), the highest power of \( x \). This gives:\[\frac{\frac{\pi}{x} + 3}{\pi - \frac{3}{x}}\]

Step 3: Evaluate the Limits term by term

As \( x \rightarrow \infty \), the terms \( \frac{\pi}{x} \) and \( \frac{3}{x} \) both approach \( 0 \). Therefore, the function simplifies to:\[ \frac{0 + 3}{\pi - 0} = \frac{3}{\pi}\]

Step 4: Write the Final Answer

The limit of the function \( f(x) \) as \( x \rightarrow \infty \) is:\[\lim_{{x \to \infty}} \frac{\pi + 3x}{\pi x - 3} = \frac{3}{\pi}\]

Key Concepts

Understanding Infinity in Limits

Infinity is a fascinating concept in calculus. When we evaluate a limit as a variable approaches infinity, we're looking at its behavior as it grows without bound. In the context of calculus, infinity allows us to make predictions about how functions behave during extreme conditions, like growing very large or shrinking to very small values.

• Infinity doesn't behave like a typical number. You can't add, subtract, multiply, or divide infinity in the same way you can with real numbers.
• In limits, when we say a variable approaches infinity, like in the expression \(x \rightarrow \infty\), we consider how the expressions evolve.
• The most common scenarios involve determining the dominant terms, especially in rational functions, to ascertain how they influence the outcome of the limits.

Thinking about limits as \(x\) approaches infinity helps identify the long-term trends in functions and can simplify complex expressions by focusing on what dominates as \(x\) becomes extremely large.

The Role of Algebraic Manipulation

Algebraic manipulation is essential when dealing with limits, especially when approaching infinity. It involves restructuring expressions to make them easier to evaluate. Here’s how it's done:

• Identify the highest power of \(x\) in both the numerator and the denominator of the function.
• Divide each term by the highest power to simplify the expression and clarify the dominant behavior of the function.
• In our example, \( f(x) = \frac{\pi + 3x}{\pi x - 3} \), the highest power of \(x\) is 1. We simplify by dividing each term by \(x\).

By dividing the entire expression by the highest power of \(x\), we approach a form where as \(x\) grows large, simpler terms tend to zero, making it clearer which terms dictate the function's behavior. This simplification shows the expression's true nature as \(x\) tends to infinity.

Steps in Limit Evaluation

Evaluating limits, especially as they approach infinity, requires a systematic approach. Let's break it down:

• Step 1: Identify influential terms. By recognizing the dominating factors in both numerator and denominator, we set the foundation for simplification.
• Step 2: Use algebraic manipulation to simplify the function, as seen when we divided terms by the highest power of \(x\).
• Step 3: Evaluate limits term by term after simplification. As extremely small or large portions shrink to zero due to division, what remains reveals the limit.

In our case, when \(x\) is infinitely large, terms like \(\frac{\pi}{x}\) and \(\frac{3}{x}\) trend to zero. The remaining terms, \(\frac{3}{\pi}\), showcase the function’s behavior in the limit. This methodical approach helps dissect complex functions and find their long-term tendencies, providing a clearer understanding of their full scope.