Determine Whether The Function $\[ F(x) = \begin{cases} \frac{1}{x}, & X \leq 2 \\ 9x - 5, & X \ \textgreater \ 2 \end{cases} \\]is Continuous At \[$ X = 2 \$\]. The Function Is $\square$ (Click For List) At \[$ X =

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Introduction

In mathematics, a function is considered continuous at a point if its graph can be drawn without lifting the pencil from the paper. In other words, a function is continuous at a point if its left-hand limit, right-hand limit, and function value at that point are all equal. In this article, we will determine whether the given function is continuous at x = 2.

The Function

The given function is defined as:

f(x)={1x,x≤29x−5,x \textgreater 2f(x) = \begin{cases} \frac{1}{x}, & x \leq 2 \\ 9x - 5, & x \ \textgreater \ 2 \end{cases}

This function has two different definitions for x ≤ 2 and x > 2. We need to check if the function is continuous at x = 2.

Left-Hand Limit

To find the left-hand limit of the function at x = 2, we need to evaluate the limit of the function as x approaches 2 from the left.

lim⁡x→2−f(x)=lim⁡x→2−1x\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} \frac{1}{x}

As x approaches 2 from the left, the value of 1/x approaches infinity. Therefore, the left-hand limit of the function at x = 2 does not exist.

Right-Hand Limit

To find the right-hand limit of the function at x = 2, we need to evaluate the limit of the function as x approaches 2 from the right.

lim⁡x→2+f(x)=lim⁡x→2+(9x−5)\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (9x - 5)

As x approaches 2 from the right, the value of 9x - 5 approaches 17. Therefore, the right-hand limit of the function at x = 2 exists and is equal to 17.

Function Value

To find the function value at x = 2, we need to evaluate the function at x = 2.

f(2)=9(2)−5=18−5=13f(2) = 9(2) - 5 = 18 - 5 = 13

Conclusion

Since the left-hand limit of the function at x = 2 does not exist, the function is not continuous at x = 2.

Is the Function Continuous at x = 2?

The function is not continuous at x = 2.

Why is the Function Not Continuous at x = 2?

The function is not continuous at x = 2 because the left-hand limit of the function at x = 2 does not exist. The left-hand limit of the function at x = 2 approaches infinity, while the right-hand limit of the function at x = 2 exists and is equal to 17. Therefore, the function is not continuous at x = 2.

What is the Conclusion?

The conclusion is that the function is not continuous at x = 2.

Why is this Important?

This is important because continuity is a fundamental property of functions in mathematics. If a function is not continuous at a point, it can have serious consequences in many areas of mathematics, such as calculus and analysis.

What are the Implications?

The implications of this result are that the function cannot be differentiated at x = 2, and the function cannot be integrated at x = 2.

What are the Next Steps?

The next steps are to investigate the behavior of the function at x = 2 in more detail, and to explore the implications of this result in different areas of mathematics.

References

  • [1] Calculus, 3rd edition, by Michael Spivak
  • [2] Real Analysis, 2nd edition, by Richard R. Goldberg

Discussion

This article has discussed the continuity of the function at x = 2. The function is not continuous at x = 2 because the left-hand limit of the function at x = 2 does not exist. The implications of this result are that the function cannot be differentiated at x = 2, and the function cannot be integrated at x = 2. The next steps are to investigate the behavior of the function at x = 2 in more detail, and to explore the implications of this result in different areas of mathematics.

Conclusion

Q: What is continuity in mathematics?

A: Continuity in mathematics refers to the property of a function that allows it to be drawn without lifting the pencil from the paper. In other words, a function is continuous at a point if its left-hand limit, right-hand limit, and function value at that point are all equal.

Q: Why is continuity important in mathematics?

A: Continuity is important in mathematics because it allows us to perform various mathematical operations, such as differentiation and integration, at a point. If a function is not continuous at a point, it can have serious consequences in many areas of mathematics.

Q: What is the left-hand limit of a function?

A: The left-hand limit of a function at a point is the limit of the function as x approaches that point from the left. In other words, it is the value that the function approaches as x gets arbitrarily close to the point from the left.

Q: What is the right-hand limit of a function?

A: The right-hand limit of a function at a point is the limit of the function as x approaches that point from the right. In other words, it is the value that the function approaches as x gets arbitrarily close to the point from the right.

Q: How do you determine if a function is continuous at a point?

A: To determine if a function is continuous at a point, you need to check if the left-hand limit, right-hand limit, and function value at that point are all equal. If they are equal, then the function is continuous at that point.

Q: What is the function value at a point?

A: The function value at a point is the value of the function at that point. In other words, it is the value that the function takes on at that point.

Q: Why is the function not continuous at x = 2?

A: The function is not continuous at x = 2 because the left-hand limit of the function at x = 2 does not exist. The left-hand limit of the function at x = 2 approaches infinity, while the right-hand limit of the function at x = 2 exists and is equal to 17.

Q: What are the implications of the function not being continuous at x = 2?

A: The implications of the function not being continuous at x = 2 are that the function cannot be differentiated at x = 2, and the function cannot be integrated at x = 2.

Q: What are the next steps in investigating the behavior of the function at x = 2?

A: The next steps in investigating the behavior of the function at x = 2 are to explore the implications of this result in different areas of mathematics, and to investigate the behavior of the function at x = 2 in more detail.

Q: What are some real-world applications of continuity in mathematics?

A: Some real-world applications of continuity in mathematics include:

  • Physics: Continuity is used to describe the motion of objects in physics. For example, the position of an object as a function of time is continuous.
  • Engineering: Continuity is used to design and analyze systems in engineering. For example, the flow of fluid through a pipe is continuous.
  • Economics: Continuity is used to model economic systems. For example, the price of a commodity as a function of time is continuous.

Q: What are some common mistakes to avoid when working with continuity in mathematics?

A: Some common mistakes to avoid when working with continuity in mathematics include:

  • Not checking the left-hand and right-hand limits: Failing to check the left-hand and right-hand limits of a function can lead to incorrect conclusions about its continuity.
  • Not checking the function value: Failing to check the function value at a point can lead to incorrect conclusions about its continuity.
  • Not considering the implications of discontinuity: Failing to consider the implications of discontinuity can lead to incorrect conclusions about the behavior of a function.