Find The Value Of $5 \lim _{x \rightarrow 9^{+}}\lfloor X\rfloor - 3 \lim _{x \rightarrow 3^{-}}\lfloor X\rfloor$, Where $\lfloor X\rfloor$ Denotes The Greatest Integer Less Than Or Equal To $x$.

Introduction to the Floor Function

The floor function, denoted by $\lfloor x\rfloor$, is a mathematical function that returns the greatest integer less than or equal to a given real number $x$. This function is also known as the greatest integer function or the integer part function. For example, $\lfloor 3.7\rfloor = 3$, $\lfloor -2.3\rfloor = -3$, and $\lfloor 5\rfloor = 5$. The floor function is an important concept in mathematics, particularly in calculus, where it is used to study the behavior of functions at the endpoints of intervals.

Limits in Calculus

A limit in calculus is a value that a function approaches as the input or independent variable approaches a certain value. In other words, a limit is the value that a function gets arbitrarily close to as the input gets arbitrarily close to a certain point. Limits are used to study the behavior of functions at the endpoints of intervals and to determine whether a function is continuous or discontinuous at a given point. There are two types of limits: one-sided limits and two-sided limits. One-sided limits are used to study the behavior of a function as the input approaches a certain value from one side, while two-sided limits are used to study the behavior of a function as the input approaches a certain value from both sides.

Evaluating the Given Expression

The given expression is $5 \lim _{x \rightarrow 9^{+}}\lfloor x\rfloor - 3 \lim _{x \rightarrow 3^{-}}\lfloor x\rfloor$. To evaluate this expression, we need to find the value of the floor function as $x$ approaches $9$ from the right and as $x$ approaches $3$ from the left. Let's start by evaluating the first limit.

Evaluating the First Limit

As $x$ approaches $9$ from the right, the floor function $\lfloor x\rfloor$ approaches $9$. This is because the greatest integer less than or equal to any number greater than $9$ is $9$. Therefore, the first limit is equal to $9$.

Evaluating the Second Limit

As $x$ approaches $3$ from the left, the floor function $\lfloor x\rfloor$ approaches $3$. This is because the greatest integer less than or equal to any number less than $3$ is $3$. Therefore, the second limit is equal to $3$.

Substituting the Values of the Limits

Now that we have found the values of the two limits, we can substitute them into the given expression. We get:

$5 \lim _{x \rightarrow 9^{+}}\lfloor x\rfloor - 3 \lim _{x \rightarrow 3^{-}}\lfloor x\rfloor = 5(9) - 3(3)$

Simplifying the Expression

Now we can simplify the expression by multiplying the numbers:

$5(9) - 3(3) = 45 - 9$

Finding the Final Answer

Finally, we can find the final answer by subtracting $9$ from $45$:

$45 - 9 = 36$

Therefore, the value of the given expression is $\boxed{36}$.

Conclusion

In this article, we have evaluated the expression $5 \lim _{x \rightarrow 9^{+}}\lfloor x\rfloor - 3 \lim _{x \rightarrow 3^{-}}\lfloor x\rfloor$ using the concept of limits and the floor function. We have found that the value of the expression is $36$. This result demonstrates the importance of understanding the floor function and limits in calculus, as they are used to study the behavior of functions at the endpoints of intervals.

Introduction

In our previous article, we evaluated the expression $5 \lim _{x \rightarrow 9^{+}}\lfloor x\rfloor - 3 \lim _{x \rightarrow 3^{-}}\lfloor x\rfloor$ using the concept of limits and the floor function. In this article, we will answer some frequently asked questions related to the floor function and limits in calculus.

Q: What is the floor function?

A: The floor function, denoted by $\lfloor x\rfloor$, is a mathematical function that returns the greatest integer less than or equal to a given real number $x$. For example, $\lfloor 3.7\rfloor = 3$, $\lfloor -2.3\rfloor = -3$, and $\lfloor 5\rfloor = 5$.

Q: What is the difference between the floor function and the ceiling function?

A: The ceiling function, denoted by $\lceil x\rceil$, is a mathematical function that returns the smallest integer greater than or equal to a given real number $x$. For example, $\lceil 3.7\rceil = 4$, $\lceil -2.3\rceil = -2$, and $\lceil 5\rceil = 5$. The main difference between the floor function and the ceiling function is that the floor function returns the greatest integer less than or equal to $x$, while the ceiling function returns the smallest integer greater than or equal to $x$.

Q: What is a limit in calculus?

A: A limit in calculus is a value that a function approaches as the input or independent variable approaches a certain value. In other words, a limit is the value that a function gets arbitrarily close to as the input gets arbitrarily close to a certain point.

Q: What is the difference between a one-sided limit and a two-sided limit?

A: A one-sided limit is a limit that is approached from one side, either from the left or from the right. For example, $\lim _{x \rightarrow 3^{-}} f(x)$ is a one-sided limit that is approached from the left, while $\lim _{x \rightarrow 3^{+}} f(x)$ is a one-sided limit that is approached from the right. A two-sided limit is a limit that is approached from both sides, and is denoted by $\lim _{x \rightarrow 3} f(x)$.

Q: How do you evaluate a limit?

A: To evaluate a limit, you need to determine the value that the function approaches as the input gets arbitrarily close to a certain point. This can be done using various techniques, such as substitution, factoring, and the squeeze theorem.

Q: What is the squeeze theorem?

A: The squeeze theorem is a mathematical theorem that states that if a function $f(x)$ is sandwiched between two other functions $g(x)$ and $h(x)$, and if the limits of $g(x)$ and $h(x)$ as $x$ approaches a certain point are equal, then the limit of $f(x)$ as $x$ approaches that point is also equal.

Q: How do you use the squeeze theorem to evaluate a limit?

A: To use the squeeze theorem to evaluate a limit, you need to find two functions $g(x)$ and $h(x)$ that sandwich the function $f(x)$, and then determine the limits of $g(x)$ and $h(x)$ as $x$ approaches a certain point. If the limits of $g(x)$ and $h(x)$ are equal, then the limit of $f(x)$ is also equal.

Conclusion

In this article, we have answered some frequently asked questions related to the floor function and limits in calculus. We hope that this article has provided a better understanding of these concepts and has helped to clarify any confusion. If you have any further questions, please don't hesitate to ask.