Two-Path Test: Understanding Variable Function Limits

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In summary, the conversation discusses the concept of limits of several variable functions and how they depend on the chosen path. It is mentioned that if the limit depends on the path chosen, then it does not exist. However, there are cases where the limit can be easily calculated and it depends on the path chosen. The problem arises when there is a contradiction, as in the given example where the limit is known to exist but the path chosen leads to a different result. The question is raised whether any important conditions were missed in this scenario and how the path test can be applied correctly. DonAntonio provides insight by explaining that the chosen path must make sense when approaching the desired point, and points out the mistake in the given example. Overall, the
  • #1
assed
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Well, I've been calculating some limits of several variable functions and got confused with something: everybody knows that if taking a path the limit depends on the path chosen then the limit doesn't exist. But, if you consider the limit

lim (x,y) -> (0,0) [itex]\frac{y^{2}x}{x^{2}+y^{2}}[/itex]

that can be easily calculated( function that converges to 0 multiplied by limited function ). ALthough, if you choose the path

y=[itex]\sqrt{\frac{c}{x-c}}x[/itex]

you'll end up with a limit equals to c, in other words, it depends oof the path you choose what would be enough for us to say the limit doesn't exist. The problem is we know it exists and my question is: did i forget to consider any important hypothesis or condition? How can the path test be applied correctly then?
I appreciate your attention.
 
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  • #2
assed said:
Well, I've been calculating some limits of several variable functions and got confused with something: everybody knows that if taking a path the limit depends on the path chosen then the limit doesn't exist. But, if you consider the limit

lim (x,y) -> (0,0) [itex]\frac{y^{2}x}{x^{2}+y^{2}}[/itex]

that can be easily calculated( function that converges to 0 multiplied by limited function ). ALthough, if you choose the path

y=[itex]\sqrt{\frac{c}{x-c}}x[/itex]

you'll end up with a limit equals to c, in other words, it depends oof the path you choose what would be enough for us to say the limit doesn't exist. The problem is we know it exists and my question is: did i forget to consider any important hypothesis or condition? How can the path test be applied correctly then?
I appreciate your attention.



The path must be chosen as to make sense when the variable(s) approach the wanted point. When [itex]\,x\to 0\,[/itex] , the expression
[tex]\sqrt{\frac{c}{x-c}}\,[/tex]
becomes undefined (over the reals) as the fraction into the square root becomes negative, no matter what number [itex]\,c\neq 0\,[/itex] is.

DonAntonio
 
  • #3
Thanks DonAntonio.
It's so obvious yet i didn't noticed it.
 
  • #4
assed said:
Thanks DonAntonio.
It's so obvious yet i didn't noticed it.


Don't worry about it (it occurs to us all, no matter whether you're a beginning undergraduate or a graduate one.

Just learn this lesson and try to remeber it in the future...where new mistakes await for us all, too. :)

DonAntonio
 

FAQ: Two-Path Test: Understanding Variable Function Limits

What is a two-path test?

A two-path test is a method used to determine the limit of a function at a specific point. It involves evaluating the limit using two different paths or approaches to the point from both sides, and if the results are the same, then the limit exists. This test is commonly used when the limit is not immediately obvious or when other methods fail.

Why is understanding variable function limits important?

Understanding variable function limits is important because it allows us to describe the behavior of a function as the input approaches a specific value. This can help us determine if the function has a specific value at that point or if it approaches a certain value as the input gets closer and closer to that point. It also has important applications in calculus and other areas of mathematics and science.

How do you perform a two-path test?

To perform a two-path test, you first need to identify the point at which you want to evaluate the limit. Then, you choose two different paths or approaches to that point, one from the left and one from the right. You then evaluate the limit using both paths and compare the results. If they are the same, then the limit exists and is equal to that value. If the results are different, then the limit does not exist.

What are the limitations of the two-path test?

One limitation of the two-path test is that it can only be used to determine the existence of a limit, not the actual value of the limit. It also requires that the two paths used are the only possible approaches to the point, which may not always be the case. Additionally, the test may not work for more complex or undefined functions.

Are there any alternative methods to the two-path test?

Yes, there are other methods for evaluating limits, such as direct substitution, algebraic manipulation, and the squeeze theorem. These methods may be more appropriate in certain situations and can sometimes provide a faster and more direct approach to determining a limit. It is important to be familiar with multiple methods for evaluating limits in order to choose the most appropriate one for a given function.

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