I\'m having trouble grasping the concepts of continuity, especially when using t

Question

I'm having trouble grasping the concepts of continuity, especially when using the two-path test. I know how to use two path-test, but i dont understand why it works.

A quick summary from what i understand of it, it bassically states that if the limit of a function approaches the same value along different paths, then it exists, if it approaches different values, then the limit does not exist. What perplexes me however is the selection of the paths, i can find paths that do not result in an equal limit for functions that do exist, which completly baffles me.

For example the function f(x,y) = 1- |xy|. So the lim(x,y) ->(0,0) f(x,y) = f(0,0) = 1, therefore it exists. However if i use the two path test, so set x = y2, the limit will equal 1, but if i set x= 1/y, i end up with 1 - (y/y) which is 1-1 = 0. So my two path test has proven my limit does not exist? im so confused it hurts

Explanation / Answer

I'm having trouble grasping the concepts of continuity, especially when using the two-path test. I know how to use two path-test, but i dont understand why it works.

Understanding what you don't understand is the most important step of learning something new :)  

A quick summary from what i understand of it, it bassically states that if the limit of a function approaches the same value along different paths, then it exists, if it approaches different values, then the limit does not exist.

This is 100% correct (well, except that you misspelled basically :P)

What perplexes me however is the selection of the paths, i can find paths that do not result in an equal limit for functions that do exist, which completly baffles me.

Let's take a look at an example:

For example the function f(x,y) = 1- |xy|. So the lim(x,y) ->(0,0) f(x,y) = f(0,0) = 1, therefore it exists.

Yep.

However if i use the two path test, so set x = y2, the limit will equal 1, but if i set x= 1/y, i end up with 1 - (y/y) which is 1-1 = 0. So my two path test has proven my limit does not exist? im so confused it hurts

The problem - just as you point out in your question - is that you're having trouble selecting paths.

Now, to show that a limit exists we would technically need to check an infinite number of paths and verify that the function is approaching the same value regardless of the path we are using to approach the point.

However, this isn't necessary. Borrowing from a fundamental idea of calculus - limits - we know that if we know that a function is continuous at a point then all we need to do to take the limit of the function at that point is to plug the point into the function.

Read that again: If we know that 1 - |xy| is continuous, we only need to take a limit at those values.

Taking the path x = y^2 is perfect. However, when you set x = 1/y, you're taking the path at the ONE point, (0,0), where the function is not continuous. That's why you're running into problems - you need to take paths where the function is continuous.

Although (0,0) is indeed the limit, the function is NOT continuous at this point - therefore, the two-path test DOES NOT WORK.

The two-path test only works for testing limits on which the function is continuous.

I hope that you found this answer useful towards your studies. It took a considerable amount of thought, time, and effort to compose, and and I'd sincerely appreciate a lifesaver rating! It would really make my day, and will allow me to continue answering your questions :)

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