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Chapter 6: Problem 9

Evaluate each limit (or state that it does not exist). $$ \lim _{x \rightarrow-\infty} \frac{1}{x^{3}} $$

Short Answer

Expert verified
The limit is 0.

Step by step solution

01

Understanding the Function

We have the function \( f(x) = \frac{1}{x^3} \). As \( x \) approaches \(-\infty\), the term \( x^3 \) becomes very large in magnitude and negative in value.
02

Analyzing the Dominant Term

Since \( x^3 \) dominates in the denominator and grows without bound as \( x \rightarrow -\infty \), the fraction \( \frac{1}{x^3} \) approaches zero.
03

Concluding the Limit

No terms in the numerator change this behavior, so as \( x \rightarrow -\infty \), \( \frac{1}{x^3} ightarrow 0 \). Thus, the limit of \( f(x) \) as \( x \) approaches \(-\infty\) is 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Evaluating Limits
In calculus, evaluating limits is about understanding the behavior of a function as the input approaches a specific value. For instance, in the expression \( \lim_{x \rightarrow -\infty} \frac{1}{x^3} \), we are interested in finding out what happens to \( \frac{1}{x^3} \) when \( x \) becomes infinitely large in the negative direction.

To evaluate such limits, follow these steps:
  • Identify the direction in which \( x \) is approaching. In this case, it is approaching \(-\infty\).
  • Consider the overall form of the function. Here, \( \frac{1}{x^3} \) is a rational function with the variable \( x \) in the denominator.
  • Analyze how each part of the function changes as \( x \) moves towards the target direction. For our specific function, \( x^3 \) in the denominator increases in magnitude, dominating the behavior as \( x \) approaches \(-\infty\).
By following these steps, you can predict the limit of a function even in complex scenarios.
Dominant Term Analysis
Dominant term analysis simplifies the process of evaluating limits by focusing on the most influential part of a function. This is particularly useful when one term of the function grows much quicker than the others, as it will dictate the function's behavior.

In our example \( \frac{1}{x^3} \), the term \( x^3 \) in the denominator becomes the dominant term as \( x \rightarrow -\infty \). Here's why:
  • As \( x \) takes on large negative values, \( x^3 \) becomes very large negatively, overwhelming any effects from other terms, especially if the numerator is constant or lower order.
  • Due to the rapidly increasing size of \( x^3 \), the value of \( \frac{1}{x^3} \) shrinks towards zero, since dividing by a huge number (in magnitude) yields a very small number.
This analysis helps in understanding that the behavior of the denominator directly results in the function approaching zero. It's a powerful technique in calculus when the limit involves functions with terms of significantly varying degrees.
Limits at Infinity
Limits at infinity explore what happens to a function as the input either grows forever in the positive or negative direction. It provides insights into the end behavior of functions.

Consider the limit \( \lim_{x \rightarrow -\infty} \frac{1}{x^3} \). As \( x \) becomes more and more negative, we look to see what happens to \( \frac{1}{x^3} \). With limits at infinity:
  • The behavior of the dominant term (in this case, \( x^3 \)) as \( x \rightarrow -\infty \) is key to understanding the function's behavior.
  • The function \( \frac{1}{x^3} \), which has a denominator growing very large in magnitude but negative, approaches zero. This is due to the magnitude of \( x^3 \) becoming overwhelming compared to the numerator.
  • Such limits can help determine horizontal asymptotes or the overall end behavior of functions.
Mastering limits at infinity allows students to predict how whether functions stabilize or tend towards particular values as inputs move towards positive or negative infinity.

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