Who is correct?

Steven G

Steven G

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My friend and I disagree on the solution to a problem. Please tell me who is correct.

Suppose g(x,y)--> L (some real number) as (x,y)-->(a,b)

Now consider g(x,y)(x-a)/(x-a) as (x,y)-->(a,b). I claim this limit is still L while my friend say it is NOT since if you approach (a,b) along the line x=a the limit is not 1/4.

I think he is wrong for a number of reasons
 
1) x = a is NOT in the Domain. Don't approach along that line.
2) Rational Function Argument: g(x,y)(x-a)/(x-a) is EXACTLY the same as g(x,y) EXCEPT at x = a. Since x = a is not the Domain, this cannot change the limit.
 
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Jomo said:
My friend and I disagree on the solution to a problem. Please tell me who is correct.

Suppose g(x,y)--> L (some real number) as (x,y)-->(a,b)

Now consider g(x,y)(x-a)/(x-a) as (x,y)-->(a,b). I claim this limit is still L while my friend say it is NOT since if you approach (a,b) along the line x=a the limit is not 1/4.

I think he is wrong for a number of reasons
Click to expand...
It isn't clear what 1/4 has to do with this question. I think I see what may be bothering your friend. The fact that [MATH]\frac {g(x,y)(x-a)}{(x-a)}[/MATH] and [MATH]g(x,y)[/MATH] don't have the same domain if [MATH](a,y) \in \text{Dom}(f)[/MATH] for [MATH]y[/MATH] in a neighborhood of [MATH]b[/MATH] means that the allowable paths for approaching [MATH](a,b)[/MATH] may not be the same. Your friend may think that means the limits aren't the same. But, as you know, that doesn't change the limit.
 
LCKurtz said:
It isn't clear what 1/4 has to do with this question. I think I see what may be bothering your friend. The fact that [MATH]\frac {g(x,y)(x-a)}{(x-a)}[/MATH] and [MATH]g(x,y)[/MATH] don't have the same domain if [MATH](a,y) \in \text{Dom}(f)[/MATH] for [MATH]y[/MATH] in a neighborhood of [MATH]b[/MATH] means that the allowable paths for approaching [MATH](a,b)[/MATH] may not be the same. Your friend may think that means the limits aren't the same. But, as you know, that doesn't change the limit.
Click to expand...
Yeah, 1/4 was the actual limit to the problem, ie I meant to write L instead of 1/4. Thanks for pointing this out.
 
tkhunny said:
1) x = a is NOT in the Domain. Don't approach along that line.
2) Rational Function Argument: g(x,y)(x-a)/(x-a) is EXACTLY the same as g(x,y) EXCEPT at x = a. Since x = a is not the Domain, this cannot change the limit.
Click to expand...
I 100% agree with you. I just wanted to show my friend that others agree with me. Thanks!
 
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