Constructing a Function g: R2→R Limiting x→a but not Limiting ||x||→||a||

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In summary, the conversation discusses the task of constructing an example where the limit of g(x) exists as x approaches a, but the limit of g(x) does not exist as the norm of x approaches the norm of a. The participants explore the idea of a continuous function g(x) that is not constant and cannot be written as a function of r = √(x^2+y^2 ). They suggest that the simplest non-constant function could potentially serve as an example.
  • #1
renolovexoxo
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I hope this is in the right place, it feels like calculus, but it's the last part of my analysis problem.

Construct an example where g: R2->R lim x->a g(x) exists but lim ||x||->||a|| g(x) does not exist

I'm having a very hard time coming up with something to put this together. I think this is my theory behind it, does anyone have any ideas on something that would work?

Specify a continuous functiong(x ⃗ )= g(x,y) on R^2, which is not constant, and which cannot be strictly written as a function of r = √(x^2+y^2 ). Then, the limit of g(x ⃗ ) as x ⃗ approaches a ⃗,a constant vector,will exist (and will equal g(a ⃗ ) ), but the limit of g(x ⃗ ) as |x ⃗ |approaches |a ⃗ | (a constant positive number) will not exist because x ⃗ can approach many different values in R2 (and still have|x ⃗ |approach |a ⃗ |), but the values that g(x ⃗ ) approach will be different.
 
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  • #2
Apart from constant functions, basically everything not too complicated works.
Write down the easiest non-constant function you can imagine, chances are good that it is an example you can use.
 

1. What does it mean to construct a function g: R2→R?

Constructing a function g: R2→R means creating a mathematical rule or formula that maps a two-dimensional input (x,y) to a one-dimensional output, typically represented by the variable z. This type of function is commonly used in multivariable calculus and can have a variety of applications in fields such as physics, engineering, and economics.

2. What does it mean to take the limit as x approaches a?

Taking the limit as x approaches a means evaluating the behavior of a function g(x) as the input value x gets closer and closer to a specific number a. This can be thought of as finding the value that g(x) approaches or "approaches from both sides" as x gets infinitely close to a, without actually reaching it. The limit of a function can provide important information about its behavior and can be used to solve more complex mathematical problems.

3. Why is the limit of g(x) as x approaches a important?

The limit of g(x) as x approaches a is important because it can help us understand the behavior of the function at a specific point. It can also be used to determine the continuity of a function, which is a key concept in calculus and other areas of mathematics. Additionally, limits can help us find the derivatives of functions, which are crucial in many real-world applications.

4. Can a function have a limit as x approaches a, but not as the magnitude of x approaches the magnitude of a?

Yes, it is possible for a function to have a limit as x approaches a, but not as the magnitude of x approaches the magnitude of a. This can occur when the function has different behavior for positive and negative values of x, and the limit as x approaches a only exists for one direction. In these cases, the limit as the magnitude of x approaches the magnitude of a does not exist because the function does not approach a specific value from both sides.

5. How do I construct a function g: R2→R with a limit as x approaches a, but not as the magnitude of x approaches the magnitude of a?

To construct a function g: R2→R with a limit as x approaches a, but not as the magnitude of x approaches the magnitude of a, you can start by defining the function for positive and negative values of x separately. For example, you can define g(x,y) = x for x > 0 and g(x,y) = -x for x < 0. Then, you can set g(a,y) = 0 to ensure that the function approaches a specific value as x approaches a. This function will have a limit as x approaches a, but not as the magnitude of x approaches the magnitude of a, since the function behaves differently for positive and negative values of x.

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