If $\lim_{n \to \infty}x_n=x,$ where $x$ is any real number,
then $\lim_{n \to \infty}\left(\lfloor x_n^2\rfloor+\lfloor x_n\rfloor\right) =?,$ where $\lfloor y\rfloor$ denotes the greatest integer less than or equal to $y.$
My approach: if $x\notin \Bbb Z$ ,then $\lfloor\cdot\rfloor$ is continuous at x.
So $\lim_{n \to \infty}\left(\lfloor x_n^2\rfloor+\lfloor x_n\rfloor\right)=x^2+x$
Now if $x\in\Bbb Z$ then
Case 1: if ${x_n}$ takes value from the left side then, $\lfloor x_n\rfloor$ converges to $x-1$
So in this case $\lim_{n \to \infty}\left(\lfloor x_n^2\rfloor+\lfloor x_n\rfloor\right)=x^2+x-1$
Case 2: if $x_n$ takes value from the right side then, $[x_n]$ converges to $x$
So in this case $\lim_{n \to \infty}\left(\lfloor x_n^2\rfloor+[x_n\rfloor\right)=x^2+x$
Is my approach correct?