$$\DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}{\mbox{#1}}$$

Here we will work on a limit from Multivariable Calculus class:
$\lim_{(x,y)\rightarrow (0,0)} \frac{(x-y)^2}{x^2+y^2}.$
by trying the limit along several paths to see that we don't get the same
value along all paths to the origin

 (%i1) f(x,y):=(x-y)^2/(x^2+y^2);
$\tag{\%{}o1}\label{o1} \operatorname{f}\left( x,y\right) :=\frac{{{\left( x-y\right) }^{2}}}{{{x}^{2}}+{{y}^{2}}}$

What happens when we try to plug in the origin?

 (%i2) f(0,0);
$\mbox{}\\\mbox{expt: undefined: 0 to a negative exponent.}\mbox{}\\\mbox{\ensuremath{\neq}0: f(x=0,y=0)}\mbox{ -- an error. To debug this try: debugmode(true);}$

Let's try the limit along the x axis, i.e. y=0:

 (%i3) limit(f(x,0),x,0);
$\tag{\%{}o3}\label{o3} 1$

And along the y axis, i.e. x=0

 (%i4) limit(f(0,y),y,0);
$\tag{\%{}o4}\label{o4} 1$

Now along the line y=x:

 (%i5) limit(f(x,x),x,0);
$\tag{\%{}o5}\label{o5} 0$

And we conclude the limit doesn't exist.
Let's take a look at the graph and notice the behavior

 (%i6) wxplot3d(f(x,y),[x,-2,2],[y,-2,2]);
$\tag{\%{}t6}\label{t6}$ $\tag{\%{}o6}\label{o6}$

This suggests something interest also along the line y=-x:

 (%i7) limit(f(x,-x),x,0);
$\tag{\%{}o7}\label{o7} 2$
Created with wxMaxima.