Multivariable limits - path problem

In summary, the conversation discusses how to prove that the limit doesn't exist by using different values for y and x. It is suggested to use y=m(x-1) and to factorize the denominator as (x-1)(x+3). The conversation also mentions the use of sin(y) and how it can lead to different constant values for the ratio.
  • #1
WonderKitten
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TL;DR Summary
What path could I use to prove that the following limit doesn't exist.
Hey, so I have the following problem:
1605213504734.png

I'm trying to prove that the limit doesn't exist (although I'm not sure if it does or not) so:
along y=mx -> x=y/m:
1605213873389.png
, which is 0 for all k≠0.
along y^n it's the same and I'm not sure what I should do next. Could I set x = sin(y)?
If I can, then the limit in that instance would be infinite, thus proving that the limit doesn't exist, right?
 
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  • #2
You cannot use ##y=mx## since it is inconsistent with ##y\to 0## and ##x\to 1##. You might try ##y=m(x-1)##.
 
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  • #3
I also suggest along @mathman advice to factorize the denominator as ##(x-1)(x+3)##.
 
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  • #4
You know that for ##y## restricted to ##-1<y<1## and ##x## sufficiently near 1 (edited, was 0, which is wrong), there are values of ##y## where ##\sin(y) = x^2+2x-3## and other values for ##y## where ##\sin(y) = 2(x^2+2x-3)##. Those ##y## values go to zero as ##x## goes to 1 and give two different constant values, 1 and 2, for the ratio.

EDIT: I had x going to the wrong value, 0, when it should have been going to 1.
 
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Related to Multivariable limits - path problem

1. What is a multivariable limit?

A multivariable limit is a mathematical concept that describes the behavior of a function at a specific point in multiple dimensions. It involves evaluating the limit of a function as it approaches a certain point from different paths.

2. Why is it important to understand multivariable limits?

Understanding multivariable limits is important because it allows us to analyze the behavior of a function in multiple dimensions. This can help us determine the continuity, differentiability, and critical points of a function, which are crucial in many areas of science and engineering.

3. How do you calculate a multivariable limit?

To calculate a multivariable limit, you must evaluate the limit of the function as it approaches the given point from different paths. This can be done by plugging in the coordinates of the point into the function and evaluating the resulting expression. If the limit is the same for all paths, then it exists. If not, the limit does not exist.

4. What are some common challenges when working with multivariable limits?

Some common challenges when working with multivariable limits include determining the correct paths to approach the point from, dealing with indeterminate forms, and understanding the concept of continuity in multiple dimensions. It is also important to be familiar with the properties of limits and how they apply to multivariable functions.

5. How are multivariable limits used in real-world applications?

Multivariable limits are used in various real-world applications, such as in physics, engineering, and economics. They are used to analyze the behavior of complex systems and to make predictions about their future behavior. For example, they can be used to determine the optimal path for a rocket to reach its destination or to predict the maximum profit for a company based on different variables.

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