I am trying to solve the question below.
My initial approach was to look at the limit along the x and y axis which both yield 0. Now, I moved on to look at the limit along the path of y = mx. When I did this, i noticed that I had to fix my x and substitute y for mx instead of saying x = y/m and y = mx.
When I did the latter, I ended up going in circles and it produced the original equation. My question is, why do I need to fix the x value? What is the mathematical reasoning?
Once you fix your $m\neq 0$, you should be good to go. For example, if you use $y=x$, you have $$\frac{x^2}{x^2+x^2},$$ which approaches $\frac{1}{2}$. And then using $x$-axis (which is $y=0$), as you have computed, it approaches $0$. If the limit exists, then along any path containing $(0,0)$ the limit should be equal. But we have demonstrated that it is not. So, the limit does not exist.
Note: You could also use the path $y=0$ to pair with $y=x$. The point is that there is no need to use three paths. Finding two which give different limits are enough to conclude the limit does not exist (although you may look at more than two if both give the same limit).