Choose The Set Of Correct Options Considering The Function Given Below:${ F(x) = \begin{cases} \frac{\sin X}{x} & \text{if } X \neq 0 \ 1 & \text{if } X = 0 \end{cases} }$1. { F(x) $}$ Is Not Continuous At [$ X = 0

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Introduction

In mathematics, the concept of continuity plays a vital role in understanding the behavior of functions. A function is said to be 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 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 discuss the continuity of a given function and determine the set of correct options.

The Given Function

The given function is defined as:

f(x)={sin⁡xxif x≠01if x=0{ f(x) = \begin{cases} \frac{\sin x}{x} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases} }

This function is a piecewise function, meaning it has different definitions for different intervals of the input variable x.

Continuity at x = 0

To determine if the function is continuous at x = 0, we need to check if the limit of the function as x approaches 0 is equal to the value of the function at x = 0.

Let's first find the limit of the function as x approaches 0:

lim⁡x→0sin⁡xx{ \lim_{x \to 0} \frac{\sin x}{x} }

Using the fundamental limit formula, we know that:

lim⁡x→0sin⁡xx=1{ \lim_{x \to 0} \frac{\sin x}{x} = 1 }

Now, let's compare this limit with the value of the function at x = 0:

f(0)=1{ f(0) = 1 }

Since the limit of the function as x approaches 0 is equal to the value of the function at x = 0, we can conclude that the function is continuous at x = 0.

Conclusion

Based on our analysis, we can conclude that the function is continuous at x = 0. Therefore, the correct option is:

  • The function is continuous at x = 0.

Discussion

The concept of continuity is crucial in mathematics, and it has numerous applications in various fields such as physics, engineering, and economics. Understanding the continuity of a function helps us to analyze its behavior and make predictions about its future values.

In this article, we discussed the continuity of a given function and determined the set of correct options. We found that the function is continuous at x = 0, which is a critical point in the function's definition.

References

  • [1] Thomas, G. B. (2010). Calculus and Analytic Geometry. Addison-Wesley.
  • [2] Spivak, M. (2008). Calculus. Cambridge University Press.

Further Reading

For further reading on the topic of continuity, we recommend the following resources:

Conclusion

Introduction

In our previous article, we discussed the continuity of a given function and determined the set of correct options. In this article, we will provide a Q&A section to further clarify the concept of continuity and address any questions or doubts that readers may have.

Q&A

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 the limit of the function as the input approaches that point is equal to the value of the function at that point.

Q: What are the conditions for a function to be continuous?

A: 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.

Q: What is the difference between continuity and discontinuity?

A: Continuity refers to the property of a function that allows it to be drawn without lifting the pencil from the paper, whereas discontinuity refers to the property of a function that has a gap or a break in its graph.

Q: Can a function be continuous at multiple points?

A: Yes, a function can be continuous at multiple points. In fact, a function can be continuous at all points in its domain.

Q: What is the significance of continuity in mathematics?

A: Continuity is a fundamental concept in mathematics that has numerous applications in various fields such as physics, engineering, and economics. Understanding the continuity of a function helps us to analyze its behavior and make predictions about its future values.

Q: How do I 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 limit of the function as the input approaches that point exists and is equal to the value of the function at that point.

Q: What are some common types of discontinuities?

A: Some common types of discontinuities include:

  • Jump discontinuity: A function has a jump discontinuity at a point if the limit of the function as the input approaches that point from the left is not equal to the limit of the function as the input approaches that point from the right.
  • Infinite discontinuity: A function has an infinite discontinuity at a point if the limit of the function as the input approaches that point is infinite.
  • Removable discontinuity: A function has a removable discontinuity at a point if the limit of the function as the input approaches that point exists, but is not equal to the value of the function at that point.

Q: Can a function be continuous at a point if it has a removable discontinuity?

A: No, a function cannot be continuous at a point if it has a removable discontinuity. A removable discontinuity is a type of discontinuity that can be removed by redefining the function at the point of discontinuity.

Q: How do I remove a removable discontinuity?

A: To remove a removable discontinuity, you need to redefine the function at the point of discontinuity so that the limit of the function as the input approaches that point exists and is equal to the value of the function at that point.

Conclusion

In conclusion, the concept of continuity is a fundamental concept in mathematics that has numerous applications in various fields. Understanding the continuity of a function helps us to analyze its behavior and make predictions about its future values. We hope this Q&A article has provided valuable insights into the concept of continuity and its importance in mathematics.