Walk into the same point from different directions and arrive at different answers. Then there is no limit.
Try the fourth function. Every path gives zero, and this time that really is the limit.
Switch paths and compare where each one lands.
In one variable there are two ways to approach a point: from the left and from the right. Check both, and if they agree you have a limit. In two variables there are infinitely many ways in — every curve that ends at the point is a legitimate approach — and the limit exists only if all of them agree.
That makes disproving a limit easy and proving one hard. To show a limit does not exist you need exactly two paths that disagree, and the fan of straight lines usually finds them: watch the value along each line settle to a different number depending on the slope.
The second function is the cautionary one. Every straight line through the origin gives zero, so a quick check says "limit is 0" — and it's wrong. Approach along the parabola instead and the value sits at ½. Straight lines are not enough, and no finite collection of paths ever is.
Which is why proving existence needs a different tool entirely: bound the whole function by something that goes to zero regardless of direction, and let the squeeze do the work. The heat map is a good hint — the ones with a genuine limit fade smoothly to a single colour at the origin, and the ones without stay stubbornly striped no matter how far you zoom in.