# Multivariable Function Limit by Squeeze Theorem

• Peacefulchaos
In summary, the speaker has uploaded their question and work to Google Docs as a PDF and did not follow the template provided. They are asking if it is appropriate to assume that r^2*cos^4(theta) goes to zero when finding the minimum of the denominator. The response is that the assumption is valid and that the speaker can save some work by manipulating the inequality before switching to polar coordinates.
Peacefulchaos
I have uploaded my question along with all of my work to Google Docs as a PDF, it can be found https://docs.google.com/fileview?id...1ZDktZmVlYjA4MTQxNzE2&hl=en&authkey=CL-llawP", which is why I did not follow the template provided. (I already had it in a PDF :\$)

I am curious if it is appropriate to assume that r^2*cos^4(theta) goes to zero when I am trying to find the minimum of the denominator. (You will see what I am talking about once you get to the step.) If you don't follow my work I'll be happy to explain my thought process.

Last edited by a moderator:
Your work is perfect, the assumption is valid since $$| \cos\theta | \le 1$$ and likewise for the sine function. You can save some work by working the inequality a little before switching polar.

## 1. What is the Squeeze Theorem?

The Squeeze Theorem, also known as the Sandwich Theorem, is a mathematical tool used to evaluate the limit of a function by comparing it to two other functions that are known to have the same limit at a given point.

## 2. How does the Squeeze Theorem work?

The Squeeze Theorem states that if two functions, f(x) and g(x), are "squeezing" a third function, h(x), such that f(x) ≤ h(x) ≤ g(x) for all values of x near a certain point, then the limit of h(x) at that point is also the limit of f(x) and g(x).

## 3. What is a multivariable function?

A multivariable function is a mathematical function that takes in multiple inputs and produces a single output. This is different from a single variable function, which only takes in one input and produces one output.

## 4. How is the Squeeze Theorem used to evaluate multivariable function limits?

To use the Squeeze Theorem to evaluate the limit of a multivariable function, we must first find two other functions that will "squeeze" the original function. We then evaluate the limit of these two functions and if they have the same limit at the point in question, then the limit of the original function at that point is also the same.

## 5. What are the benefits of using the Squeeze Theorem for evaluating multivariable function limits?

The Squeeze Theorem is a powerful tool for evaluating multivariable function limits because it allows us to determine the limit of a complicated function by comparing it to simpler functions. This can save time and effort in solving complex problems and can also provide a more accurate result.

• Calculus and Beyond Homework Help
Replies
6
Views
2K
• Calculus and Beyond Homework Help
Replies
8
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
2K
• Calculus and Beyond Homework Help
Replies
10
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
6K
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
1K