Two-path test proof

Question

If we want to show that the limit of function of two variables doesn't exist , it is sufficient two paths which have different limits . Intuitively , it seems correct but how we can prove it with the using of definition of limit ?

Answer 1

Denote by $D$ the domain of your function, and pick $x_0 \in D$.

If you want the classic $\varepsilon - \delta$ definition of continuity of $f$ at $x_0$, here it is: $$\forall \varepsilon >0, \ \ \ \exists \delta >0 : \ \ \ \forall x \in D, \ \ \ |x-x_0|< \delta \Rightarrow |f(x)-f(x_0)| < \varepsilon \ \ \ \ \ \ (*)$$ Let's write the condition: "for some path the limit is $L$" $$\gamma_1:[0,1] \to D \ \ \mathrm{continuous}, \ \ \ \gamma_1(1)=x_0, \ \ \ \lim_{t \to 1}f( \gamma_1 (t))=L$$

A similar condition for some second path, with limit $L' \neq L$:

$$\gamma_2:[0,1] \to D \ \ \mathrm{continuous}, \ \ \ \gamma_2(1)=x_0, \ \ \ \lim_{t \to 1}f( \gamma_2 (t))=L'$$

Now, pick $\varepsilon = \frac{1}{9}|L-L'|$, and $\delta$ accordingly to $(*)$. By continuity of $\gamma_1$ and $\gamma_2$ at $t=1$, there esists $\delta_2$ such that $$\forall t \in [0,1], \ \ |1-t|< \delta_2 \Rightarrow (|\gamma_1(t)- x_0| < \delta \ \ \ \ \wedge \ \ \ |\gamma_2(t)-x_0| < \delta)$$ Thus, by $(*)$ you have $$|f(\gamma_1(t)) - f(x_0)| < \varepsilon \ \ \ \wedge \ \ \ |f(\gamma_2(t)) - f(x_0)| < \varepsilon$$

Moreover, there exists $\delta_3 >0$ such that $$\forall t \in [0,1], \ \ |1-t|< \delta_3 \Rightarrow (|f(\gamma_1(t))- L| < \varepsilon \ \ \ \ \wedge \ \ \ |f(\gamma_2(t))-L'| < \varepsilon)$$ Finally, take any $t \in [0,1]$ with $t < \delta_2, \delta_3$. Then $$|f( \gamma_1(t)) - f( \gamma_2(t))| = |(f( \gamma_1(t)) -f(x_0)) - ( f( \gamma_2(t))- f(x_0))| \le 2 \varepsilon= \frac{2}{9}|L-L'|$$ on the other hand $$|f( \gamma_1(t)) - f( \gamma_2(t))|= |(f( \gamma_1(t))-L) - (f( \gamma_2(t))-L') + (L-L')|\ge |L-L'|- 2 \varepsilon = \\ = \frac{7}{9}|L-L'|$$ which is a contradiction.

Now, you can see that writing all of this using $\varepsilon - \delta$ becomes very heavy and tedious: this is the reason why nobody wanted to answer you in these terms.