Problem 23 - Calculus Early Transcendentals

Question

Determine the following limits. $$\lim _{x \rightarrow \infty}\left(-12 x^{-5}\right)$$

Step-by-Step Solution

Answer

Answer: The limit of the function $$-12x^{-5}$$ as x approaches infinity is 0.

Step 1: 1. Analyze the Given Function

Observe that the given function is of the form $$-12x^{-5}$$. As x approaches infinity, the term $$x^{-5}$$ will approach zero, since the exponent is negative.

Step 2: 2. Calculate the Limit

Now, we will calculate the limit of the function as x approaches infinity: $$\lim_{x \rightarrow \infty}\left(-12 x^{-5}\right)$$ As the term $$x^{-5}$$ approaches zero, the function becomes: $$\lim_{x \rightarrow \infty}\left(-12 \cdot 0\right)$$ Multiply -12 by 0, which results in 0: $$\lim_{x \rightarrow \infty}\left(0\right)$$ Therefore, as x approaches infinity, the limit of the function $$-12x^{-5}$$ is equal to 0.

Key Concepts

Infinite Limits

In the realm of calculus, understanding infinite limits is crucial. When we discuss infinite limits, we're looking at what happens to a function as the variable approaches infinity. In simpler terms, we are interested in the behavior of the function when the variable gets very, very large.

For example, in the function \(-12x^{-5}\), as \(x\) increases towards infinity, the expression \(x^{-5}\) implies that \(x\) is raised to the negative fifth power. This means that a large \(x\) value makes the expression very small. Essentially, \(x^{-5}\) approaches zero as \(x\) becomes infinitely large. Since we're multiplying this near-zero value by -12, the whole expression tends towards zero.

Here, it's vital to distinguish between concepts like a variable going to infinity versus a function's limit going to infinity. In our example, although \(x\) goes to infinity, the function itself goes to zero. This demonstrates an **infinite limit** where the input grows without bounds, affecting the behavior of the function.

Exponential Decay

Exponential decay is a fascinating yet important concept. It's observed when values decrease rapidly at first, then slowly over time. In the context of limits, this often involves terms with negative exponents.

Take \(-12x^{-5}\) from our exercise. The term \(x^{-5}\) is analogous to the effect of exponential decay because as \(x\) grows, \(x^{-5}\) diminishes. This rapid decrease towards zero is what defines exponential decay within limits. Hence, when you see a negative exponent in a function, anticipate this behavior: quick decline settling toward a minimal value.

Important characteristics of exponential decay include:

• A starting point which quickly falls off.
• Eventually tapering toward zero or some minimal value.

This process in functions with negative exponents is pivotal, ensuring you predict how terms behave as variables expand infinitely.

Evaluating Limits

The ability to evaluate limits helps us understand the behavior of functions as variables approach particular values, including infinity. Let's go step-by-step for clarity using the example \(-12x^{-5}\).

• **Recognize the form:** Here, it's crucial to identify the form, which is \(-12x^{-5}\). This guides our expectations since \(x^{-5}\) hints at exponential decay.
• **Analyze component behavior:** With \(x^{-5}\) as \(x\) approaches infinity, it moves nearly to zero. Therefore, focus on how \(x^{-5}\) changes as \(x\) gets large.
• **Multiply and conclude:** Multiply the resulting small value by -12. This shows that the entire function approaches zero, reflecting both infinite limit characteristics and exponential decay.

By understanding the pieces, we've successfully evaluated that \(-12x^{-5}\) tends to zero as \(x\) increases to infinity. Thus, the limit of the function approaches zero. This breakdown, focusing on the negative exponent and multiplication, emphasizes a structured approach to simplify complex limits.