I came across the following passage:
Limit of a function $f$ along the line $l: (x_0 +t\cos a, y_0+t\sin a)$, $t\in \mathbb{R}, a\in [0,\pi]$, through the point $(x_0, y_0)$ is the following limit: $$\lim_{t\to 0}f(x_0 +t\cos a, y_0+t\sin a)$$
I don't understand the representation of the line $l$. Is the collection of points $(x_0 +t\cos a, y_0+t\sin a)$ a set of points that satisfies the equation of the line $l$ ? I'm familiar with polar coordinates and I've tried representing a point $(x,y)$ and plugging in the equation $y=mx+n$ but I don't get anything similar to the stuff in the passage above.
As you request I try to give more details in an answer. Suppose $(x_0,y_0)\equiv (0,0)$. You are calculating then the limit
$$\lim_{(x,mx)\to(0,0)} f(x,y).$$
Therefore, as $x$ goes to $0$, so does $y$, following the line $$r: y=mx.$$
Suppose the line forms an angle $\alpha$ with the positive $x$-semiaxis. Then $r$ can be expressed in parametric form as $$ \begin{cases} x=t\cos\alpha\\ y=t \sin\alpha, \end{cases} $$ where $t$ runs through all real values, letting $(x,y)$ cover the entire line (with $t=0$ corresponding to the origin of the axes). Now your limit can be expressed as $$\lim_{t\to 0} f(t\cos\alpha,t\sin\alpha).$$