Comparing function limits

The limit fails to exist because the function approaches different values along different paths to the origin. The epsilon-delta definition requires uniform control of every nearby domain point, meaning all paths must yield the same limit.

Conditions: The function is $f(x,y)=xy/(x^2+y^2)$.; The limit is evaluated as $(x,y) \to (0,0)$.; The domain excludes the origin $(0,0)$.

Along the line $y=kx$, substituting into the function yields $f(x,kx) = \frac{k}{1+k^2}$ for $x \neq 0$. This value depends on the slope $k$ and is constant with respect to $x$.

Conditions: The function is $f(x,y)=xy/(x^2+y^2)$.; The path is the straight line $y=kx$.; $x \neq 0$.

A two-sided one-variable limit exists if the left and right limits both exist and agree, provided the domain permits approach from both sides. This consistency ensures that the function approaches the same value regardless of the direction of approach.

Conditions: The domain permits approach from both sides of the point.; The left limit $\lim_{x\to0^-}f(x)$ exists.; The right limit $\lim_{x\to0^+}f(x)$ exists.; The left and right limits are equal to the same value $A$.

The epsilon-delta definition requires that for every $\varepsilon > 0$, there exists a $\delta > 0$ such that every domain point sufficiently close to, but distinct from, the origin gives an output close to $A$. This means uniform control of every nearby domain point, rather than just checking a few paths.

Conditions: The input is a point in the plane $(x,y)$.; The limit is evaluated as $(x,y) \to (0,0)$.; The function is defined on a domain containing a punctured neighborhood of the origin.