In Buck's Advanced Calculus book, I saw an exercise that says
Let $f$ be defined by $\displaystyle{f(x,y)=\frac{x^2y^2}{x^2+y^2},}$ with $f(0,0)=0.$ By checking various sequences, test this for continuity at $(0,0).$ Can you tell whether or not it is continuous there?
If we let $x^2y^2=x^2+y^2,$ we will find that$$y^2(x^2-1)=x^2\implies y^2=\frac{x^2}{x^2-1}\implies y=\sqrt{\frac{x^2}{x^2-1}}$$and if we let $\displaystyle{p_n=(x_n,y_n)=\left(x_n,\sqrt\frac{x_n^2}{x_n^2-1}\right)}$ with $\lim x_n=0,$ $$\lim p_n=\left(0,\lim\sqrt\frac{x_n^2}{x_n^2-1}\right)=\left(0,\sqrt\frac{0}{-1}\right)=\left(0,0\right)$$ but because of $x_n^2y_n^2=x_n^2+y_n^2,$ $\lim f(p_n)$ will be equal to $1.$ Hence, $f$ is not continuous at $(0,0).$ But the book says it is. Where did my solution fail?
The expression $\sqrt{\frac{x^2}{x^2-1}}$ only makes sense if $|x|>1$. So, if $\lim_{n\to\infty}x_n=0$, it makes no sense to talk about $\sqrt{\frac{x_n^{\,2}}{x_n^{\,2}-1}}$ when $n\gg1$.