Here's a site that explains the concept of limits with multivariable functions very nicely, with examples.
Pauls Online Notes : Calculus III - Limits
Hiya, I'm having to learn about limits really quickly and am struggling to fully grasp the concept. While I understand the basic concept, I'm having trouble figuring out what can be considered as (dis-)proof of a limit, and also the Epsilon-Delta definition of a limit. I can't seem to find anything that offers an explaination of these which is not in mathematical terms - I am not a mathematician, hence the difficulty.
I have a few example functions and would appreciate any help, although I would prefer more descriptive help rather than just the maths. All functions are defined as 0 for (x,y)=(0,0) and I would like to test the continuity at (0,0).
1.
2.
3.
Here are my approaches to these functions...
Function 1.
We can rewrite the function as . My notes feature fractions with the same powers for both numerator and denominator - is this proof of a limit?
Function 2.
If we consider the result of along the line , the function becomes . As this is constant, but the function is defined as (0,0) at (x,y)=(0,0), we have a large step rather than a gradual convergence, therefore is not continuous at (0,0).
Function 3.
If we consider the function when and , the result of the function, the result will be -1 and 1 respectively. As with function 2, we have a constant value along these lines everywhere except (0,0), therefore is not continuous at (0,0).
Thank you in advance for any help or advice you can offer.
Here's a site that explains the concept of limits with multivariable functions very nicely, with examples.
Pauls Online Notes : Calculus III - Limits