At What Points Is The Given Function Continuous?${ Y = \frac{1}{|x|+9} - \frac{x^3}{3} }$The Set Of { X $}$-values Where The Function Is Continuous Is { \square$}$.(Type Your Answer In Interval Notation.)

Introduction

In calculus, the continuity of a function is a crucial concept that deals with the behavior of a function at a given point. A function is said to be continuous at a point if the limit of the function as the input approaches that point is equal to the value of the function at that point. In this article, we will explore the points at which the given function is continuous.

The Given Function

The given function is:

${ y = \frac{1}{|x|+9} - \frac{x^3}{3} }$

This function involves absolute value and polynomial terms. To determine the points of continuity, we need to analyze the behavior of the function at different intervals.

Points of Continuity

A function is continuous at a point if the following conditions are met:

1. The function is defined at the point.
2. The limit of the function as the input approaches the point exists.
3. The limit of the function as the input approaches the point is equal to the value of the function at the point.

Let's analyze the given function at different intervals to determine the points of continuity.

Interval 1: $(-\infty, -3)$

In this interval, the absolute value term $|x|$ is negative, and the function can be rewritten as:

${ y = \frac{1}{-x+9} - \frac{x^3}{3} }$

The function is defined at all points in this interval, and the limit of the function as $x$ approaches $-\infty$ exists. However, the limit of the function as $x$ approaches $-3$ does not exist due to the absolute value term.

Interval 2: $(-3, 3)$

In this interval, the absolute value term $|x|$ is non-negative, and the function can be rewritten as:

${ y = \frac{1}{x+9} - \frac{x^3}{3} }$

The function is defined at all points in this interval, and the limit of the function as $x$ approaches $3$ exists. However, the limit of the function as $x$ approaches $-3$ does not exist due to the absolute value term.

Interval 3: $(3, \infty)$

In this interval, the absolute value term $|x|$ is positive, and the function can be rewritten as:

${ y = \frac{1}{x+9} - \frac{x^3}{3} }$

The function is defined at all points in this interval, and the limit of the function as $x$ approaches $\infty$ exists.

Conclusion

Based on the analysis of the given function at different intervals, we can conclude that the function is continuous at all points except at $x = -3$ and $x = 3$. The set of $x$-values where the function is continuous is:

${ (-\infty, -3) \cup (-3, 3) \cup (3, \infty) }$

This can be written in interval notation as:

${ (-\infty, -3) \cup (-3, 3) \cup (3, \infty) }$

Final Answer

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Introduction

In our previous article, we explored the points at which the given function is continuous. In this article, we will provide a Q&A section to further clarify the concepts and provide additional insights.

Q1: What is the significance of the absolute value term in the given function?

A1: The absolute value term $|x|$ in the given function plays a crucial role in determining the points of continuity. The absolute value term changes the behavior of the function at different intervals, and it is essential to analyze its behavior to determine the points of continuity.

Q2: Why is the limit of the function as $x$ approaches $-3$ not defined?

A2: The limit of the function as $x$ approaches $-3$ is not defined because the absolute value term $|x|$ becomes zero at $x = -3$. This causes the function to become undefined at this point.

Q3: What is the behavior of the function as $x$ approaches $\infty$?

A3: As $x$ approaches $\infty$, the absolute value term $|x|$ becomes very large, and the function approaches zero. This is because the term $\frac{1}{|x|+9}$ approaches zero as $x$ approaches $\infty$.

Q4: How does the given function behave at the point $x = 3$?

A4: At the point $x = 3$, the absolute value term $|x|$ becomes zero, and the function becomes undefined. This is because the term $\frac{1}{|x|+9}$ becomes undefined at this point.

Q5: What is the set of $x$-values where the function is continuous?

A5: The set of $x$-values where the function is continuous is:

${ (-\infty, -3) \cup (-3, 3) \cup (3, \infty) }$

This can be written in interval notation as:

${ (-\infty, -3) \cup (-3, 3) \cup (3, \infty) }$

Q6: How can we determine the points of continuity of a function?

A6: To determine the points of continuity of a function, we need to analyze the behavior of the function at different intervals. We need to check if the function is defined at the point, if the limit of the function as the input approaches the point exists, and if the limit of the function as the input approaches the point is equal to the value of the function at the point.

Q7: What is the importance of continuity in calculus?

A7: Continuity is a crucial concept in calculus that deals with the behavior of a function at a given point. It is essential to understand the points of continuity of a function to analyze its behavior and make predictions about its behavior.

Q8: How can we use the given function to model real-world phenomena?

A8: The given function can be used to model real-world phenomena such as population growth, chemical reactions, and electrical circuits. The function can be used to analyze the behavior of these phenomena and make predictions about their behavior.

Conclusion

In this Q&A article, we have provided additional insights into the points of continuity of the given function. We have answered questions about the significance of the absolute value term, the behavior of the function at different intervals, and the importance of continuity in calculus. We hope that this article has provided a better understanding of the concepts and has been helpful in clarifying any doubts.