Contrapositive proof

Contrapositive proofs are analogous as the opposite of direct proofs. In other words, if the operation between $A$ and $B$ are not true, then $A$ is not true or $B$ is not true.

$$\overline{A\wedge B}=\bar{A}\vee \bar{B}$$

Use in limits

The two-path test can prove non-existence of limits. If $f(x,y)$ approaches two different values as $(x,y)$ approaches $(a,b)$, then the limit does not exist, since it must approach along all paths in the domain.

We have two options: one is to parametrise $x$ in terms of $y$ or $y$ in terms of $x$. This comes down to guesswork, but a good option is to use $y=m{x}^n$ or $x=m{y}^n$ (it doesn’t really matter). If the limit depends on $m$, then the limit DNE because it depends on the slope of the line (and thus on a particular path). But! If we still end up with two variables, we should try something else, because we want to express it in terms of one variable.

The function we use should pass through the boundary point. For example, if the denominator is $x+y-1$, then $y=mx+1$ since it passes through the boundary point.

Polar coordinates

If we express the function in terms of different variables, and determine whether it holds for all values of the new variable. The mapping to the new variables should be bijective. The easiest method is via polar coordinates, where $x=r\cos \theta$ and $y=r\sin \theta$.

An easy case to use it with is $x^2+y^2$. Though a limit at a boundary point should be examined closely: $(x,y)=(0,0)$ may correspond to $(r,\theta )=(0,\theta )$.