Limits Across Paths

${\lim }_{x\to x_0}f(x)\text{ along }c(t)={\lim }_{t\to b}f(c(t))$

Continuous Definition

• Given a Path $c:[a,b]\to {\mathbb{R}}^n$ is a curve or path.
• If ${\lim }_{t\to b}c(t)=x_0$ (continuous)
• Then, ${\lim }_{x\to x_0}f(x)$ along path $c(t)$ is $f(x_0)$

Example

Consider limit ${\lim }_{(x,y)\to (0,0)}\frac{x{y}^2}{x^3+y^3}$ along paths $c_1(t)=(0,t)$ This will be $f(c_1(t))=f(0,t)=\frac{t^2\ast 0}{t^3+0}=0$