A common belief of students in real analysis is that if $$\lim_{x\to x_0}f(x,y_0),\qquad\lim_{y\to y_0}f(x_0,y)$$ exist and are both equal to $l$, then the function has limit $l$ in $(x_0,y_0)$. It is easly to show counter-examples. More difficult is to show that also the belief $$\lim_{t\to 0}f(x_0+ht,y_0+kt)=l,\quad\forall\;(h,k)\neq(0,0)\quad\Rightarrow\quad\lim_{(x,y)\to(x_0,y_0)}f(x,y)=l$$ is false. For completeness's sake (presumably anybody who ever taught calculus has seen it, but it's easily forgotten) the standard counterexample is $$f(x,y)=\frac{xy^2}{x^2+y^4}$$ at $(0,0$).
A common belief of students in real analysis is that if $$\lim_{x\to x_0}f(x,y_0),\qquad\lim_{y\to y_0}f(x_0,y)$$ exist and are both equal to $l$, then the function has limit $l$ in $(x_0,y_0)$. It is easly to show counter-examples. More difficult is to show that also the belief $$\lim_{t\to 0}f(x_0+ht,y_0+kt)=l,\quad\forall\;(h,k)\neq(0,0)\quad\Rightarrow\quad\lim_{(x,y)\to(x_0,y_0)}f(x,y)=l$$ is false.