Problem 38 - Calculus Early Transcendentals

Question

Evaluate the following limits. $$\lim _{x \rightarrow \infty} \frac{4 x^{3}-2 x^{2}+6}{\pi x^{3}+4}$$

Step-by-Step Solution

Answer

Question: Evaluate the limit of the given rational function as x approaches infinity: $$\lim_{x\to\infty} \frac{4x^3 - 2x^2 + 6}{\pi x^3 + 4}$$ Answer: The limit as x approaches infinity is $$\frac{4}{\pi}$$.

Step 1: Divide the numerator and the denominator by the highest power of x.

We will divide every term in the numerator and denominator by \(x^{3}\): $$\frac{4x^3 - 2x^2 + 6}{\pi x^3 + 4} \cdot \frac{\frac{1}{x^3}}{\frac{1}{x^3}} = \frac{4 - \frac{2}{x} + \frac{6}{x^3}}{\pi + \frac{4}{x^3}}$$

Step 2: Evaluate the limit as x approaches infinity.

Now, as \(x \rightarrow \infty\), the terms \(\frac{2}{x}\), \(\frac{6}{x^3}\), and \(\frac{4}{x^3}\) all approach 0, because the numerator is a constant while the denominator is approaching infinity. So we have: $$\lim_{x\to\infty} \frac{4 - \frac{2}{x} + \frac{6}{x^3}}{\pi + \frac{4}{x^3}} = \frac{4 - 0 + 0}{\pi + 0} = \frac{4}{\pi}$$ So, the limit is: $$\lim_{x\to\infty} \frac{4x^3 - 2x^2 + 6}{\pi x^3 + 4} = \frac{4}{\pi}$$

Key Concepts

Evaluating Limits

When approaching the subject of limits in calculus, evaluating limits is a fundamental skill that enables students to determine the behavior of a function as its input approaches a particular value. For instance, in our exercise, we are interested in finding the value that the function \( \frac{4x^3 - 2x^2 + 6}{\pi x^3 + 4} \) approaches as \(x\rightarrow\infty \).

To tackle this, we often employ the technique of simplifying the expression by dividing the terms by the highest power of \(x\) present in the function. This method effectively reduces the complexity of the problem and allows us to easily discern the function's behavior at infinity. As \(x\) grows larger, any terms with \(x\) in the denominator tend to zero. Here, by dividing each term by \(x^3\), we isolated \(4\), a constant, in the numerator, while all other terms with \(x\) effectively vanish. This results in \( \frac{4}{\pi} \), providing us with the limit of the function as \(x\) approaches infinity. This step-by-step approach makes the problem much more manageable and transparent, helping students to see how the function behaves without having to perform laborious calculations for large values of \(x\).

Infinite Limits

The concept of infinite limits might sound abstract at first, but it's essentially about understanding how a function behaves as it approaches a very large (infinite) or very small (approaching negative infinity) value. Unlike limits that approach a finite number, an infinite limit tells us about the unbounded growth or decline of a function.

In our exercise, we deal with an \(x\) approaching infinity. At first glance, one might think that the function itself must go to infinity. However, by scrutinizing the rate at which the terms grow or diminish relative to each other, we can often ascertain that certain functions approach a finite limit, even as \(x\) grows without bound. Our step-by-step solution demonstrated that despite the powers of \(x\) involved, the leading terms in the numerator and denominator dictate the long-term behavior of the function. In general, if the highest powers of \(x\) are the same in both the numerator and denominator, we may arrive at a finite limit, just as we found the function to approach \(\frac{4}{\pi}\) as \(x \rightarrow \text{infinity}\).

Limit Laws

With a solid understanding of limit laws, students can break down complex limit problems into simpler steps. These laws are a set of rules that provide a mathematical structure to the process of finding limits. They hold the key to not just evaluating limits efficiently but also to ensuring that each step follows a logical and valid mathematical progression.

Some of the fundamental limit laws include the sum law, product law, and quotient law, all of which are evident in our step-by-step solution. For instance, when we simplify the limit expression by dividing by \(x^3\), we're using the quotient law, which tells us that the limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero. In our exercise, as the terms \(\frac{2}{x}\), \(\frac{6}{x^3}\), and \(\frac{4}{x^3}\) approach zero due to the increasing value of \(x\), we are allowed to separate these terms out following the sum law. This understanding is vital, as it justifies why certain terms can be disregarded when considering the limit of \(x \rightarrow \text{infinity}\), leading to the final, simplified result of \(\frac{4}{\pi}\). Mastering these laws equips students with the tools to approach a wide range of limit problems with confidence.