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?
A multivariable limit is a mathematical concept that describes the behavior of a function as the input values approach a specific point in a multi-dimensional space. It is used to determine if a function has a finite value or tends towards infinity as the input values approach a given point.
To calculate a multivariable limit, you must first determine the limit along each coordinate axis separately. This can be done by plugging in the coordinates of the given point into the function. If the limits along each axis are equal, then the multivariable limit exists. If they are not equal, the limit does not exist.
A one-sided multivariable limit only considers the behavior of a function as the input values approach a given point from one side. This is useful when dealing with functions that have discontinuities. A two-sided limit considers the behavior from both sides and is used to determine the overall limit at a point.
To show the existence of a multivariable limit, you must prove that the limit along each coordinate axis is equal. This can be done by using the definition of a limit and plugging in the coordinates of the given point into the function. If the resulting limits are equal, then the multivariable limit exists.
The non-existence of a multivariable limit can be determined by showing that the limit along at least one coordinate axis is not equal to the others. This can be done by using the definition of a limit and plugging in the coordinates of the given point into the function. If the resulting limits are not equal, then the multivariable limit does not exist.