Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. \(\lim _{\theta \rightarrow-\infty} \frac{\pi \theta^{5}}{\theta^{5}-5 \theta^{4}}\)

Step-by-Step Solution

Answer

The limit is \(\pi\).

Step 1: Identify Dominant Terms

First, identify the dominant terms in the numerator and the denominator. Here, both the numerator \(\pi \theta^5\) and the denominator \(\theta^5 - 5\theta^4\) have dominant terms of \(\theta^5\).

Step 2: Factor Out Dominant Terms

Factor \(\theta^5\) out of both the numerator and the denominator. This simplifies the expression to \(\frac{\pi \theta^5}{\theta^5(1 - \frac{5}{\theta})}\).

Step 3: Cancel the Common Factor

Cancel \(\theta^5\) from both the numerator and the denominator to get \(\frac{\pi}{1 - \frac{5}{\theta}}\).

Step 4: Evaluate the Limit as \(\theta \to -\infty\)

As \(\theta \to -\infty\), the term \(\frac{5}{\theta}\) approaches 0 because the denominator becomes very large making the fraction near zero. Thus, the expression simplifies to \(\frac{\pi}{1-0}\).

Key Concepts

Dominant Terms in Limits

Finding limits in calculus often starts with identifying the dominant terms within an expression. Dominant terms are those that have the most significant impact on the value of a function as the variable approaches a particular point, often infinity or negative infinity. In our exercise, both the numerator \(\pi \theta^5\) and the denominator \(\theta^5 - 5\theta^4\) share \(\theta^5\) as their dominant term. This is because, as \(\theta\) becomes very large or very negative, the highest power term tends to "dominate" the expression, determining its ultimate behavior. Recognizing these terms is crucial, as they allow us to simplify our expressions significantly when evaluating limits.

Factorization in Calculus

Factorization in calculus is a powerful tool for simplifying complex expressions, especially when finding limits. The key is to factor out the dominant term from an expression. Once identified, these dominant terms can be factored, often leading to simplification.In our example, we factor \(\theta^5\) out from both the numerator and the denominator. This results in the expression \(\frac{\pi \theta^5}{\theta^5(1 - \frac{5}{\theta})}\). By factoring out \(\theta^5\), not only do we simplify the expression, but we also set the stage for canceling common terms. This factorization step is integral as it directly impacts how the limit behaves, especially when approaching infinity or negative infinity.

Understanding Asymptotic Behavior

Asymptotic behavior describes how a function behaves as the input approaches a particular value, such as infinity. In our exercise, as \(\theta\) moves toward negative infinity, we observe the term \(\frac{5}{\theta}\) approaching zero. Large values of \(|\theta|\) make the fraction smaller because the denominator continues to grow.This simplification leaves us with \(\frac{\pi}{1 - 0}\). Once the \(\frac{5}{\theta}\) term fades, the function behaves like \(\frac{\pi}{1}\), showing that its limit is \(\pi\). By understanding asymptotic behavior, we can predict and calculate how functions behave at extreme values, ensuring a precise evaluation of limits.