Two variables limit with weird requirements [closed]

Question

I currently have a piecewise-defined function:

$$f(x,y)=\begin{cases}\frac{\sin(xy)}{xy}&\text{ if }xy\ne0\\1&\text{ if }xy=0.\end{cases}$$

and it requires me to discuss the continuity of $f(x,y)$. I found out the way to do this is to find the limit of function when $xy$ approaches $0$. However, I wonder how should I compute the limit: should I just directly replace $xy$ with some variable $m$ and say $m$ tends to $0$, so the limit is $1$ and it is continuous? Thank you.

Answer 1

I think that the natural approach is to express your function as $g\circ h$, with$$\begin{array}{rccc}h\colon&\Bbb R^2&\longrightarrow&\Bbb R\\&(x,y)&\mapsto&xy\end{array}$$and$$\begin{array}{rccc}g\colon&\Bbb R&\longrightarrow&\Bbb R\\&x&\mapsto&\begin{cases}\frac{\sin(x)}x&\text{ if }x\ne0\\1&\text{ if }x=0.\end{cases}\end{array}$$And now, since $\lim_{(x,y)\to(0,0)}h(x,y)=0$ and $\lim_{x\to0}g(x)=1$, you have $\lim_{(x,y)\to(0,0)}f(x,y)=1$.