The discussion revolves around evaluating the limit of a multivariable function as it approaches the point (1,0). The function in question is ln((1+y^2)/(x^2+xy)), and participants are exploring the conditions under which this limit exists or does not exist.
The discussion is active, with participants sharing their attempts and questioning the validity of their approaches. There is recognition of the need for a rigorous ε-δ argument to establish the limit's existence. Some participants suggest that different paths may yield different results, indicating a lack of consensus on the limit's behavior.
Participants note that the limit is being evaluated as (x,y) approaches (1,0) rather than (0,0), which may influence the methods used. There is also a reference to the potential for the limit to be undefined along certain paths, such as y=x.
I don't see that there's any difficulty as long as x→1 and y→0. ln(1/1) = 0mike1967 said:Homework Statement
lim(x,y)->(1,0) of ln((1+y^2)/(x^2+xy))
Homework Equations
The Attempt at a Solution
Used two paths,
x=1
y=0
both gave my lim=0
so I tried x=rsin y=rcos, in attempt to use ε-δ to prove it.
got to ln((1+r^2sin^2)/(r^2cos(cos+sin)))
not sure where to go from here.
Yes, what your prof. said makes sense. I'm pretty sure that the functions that cause trouble are of indeterminate form, usually 0/0 . Very often the limit is being taken as (x,y)→(0,0) in which case using polar coordinates is often a big help.mike1967 said:My issue is in lecture my professor made it clear that finding any finite number of ways a function approached the same number did not prove that the lim was equal to that, in this case 0, because there are infinite number of ways (x,y) can approach the point. Does this make sense or did I misunderstand? Basically the only way he taught us to prove a lim existed was to use the ε-δ.
The line y=x doesn't go through the point (1,0) .mike1967 said:Well along the path x=y the limit blows up, 1/0, so then the limit does not exist?