# One of the Weirdest, Most Treacherous Math Problems You Will Ever Encounter

*A limit problem conjured up from the depths of hell.*

Anyone who’s taken calculus: can you solve this math problem?

It might look simple but it’s actually one of the weirdest, most treacherous problems you’ll ever encounter.

Go ahead, give it a try before reading on.

*(a comment on my StackExchange answer introducing this problem)*

The result is NOT correct if you pretend the limit only has one variable.

Doing so evaluates the limit along the path $y=x,$ just one of many possible paths by which we could approach the point $(0,0).$

Now, if you’ve taken multivariable calc, you probably knew that already.

But did you know that if you evaluate the limit along ALL lines through the origin, you STILL get a result of 0, and that result of 0 is still INCORRECT!

Why is this still incorrect? Because we’ve only checked LINEAR paths. You get a different result if you check the NON-linear path $y=x^2.$

Our limit evaluates to 0 along all LINEAR paths, but NOT along all paths IN GENERAL.

It comes out to $\frac{1}{2}$ along the quadratic path $y=x^2.$

So, generally speaking, the answer is that THE LIMIT DOES NOT EXIST.

To better understand this, you can take a look at the graph.

It’s super weird and counterintuitive that you can approach the SAME place through all linear paths, yet approach a DIFFERENT place through a quadratic path.

But it’s true, and the graph below shows how it’s possible: