$$\lim_{(x,y) \to (0,0)} y\sin(1/x)$$
with the change of variable $y = \frac 1u$, $x = \frac 1v$, $u, v \to \infty$, we get
$$\lim_{(u,v) \to (\infty, \infty)} \frac{\sin v}{u}$$
which tends to $0$, hence the limit is $0$.
I have seen this question, which states that a similar limit exists. However, my textbook (Thomas Finney) mentions that the limit does not exist
So where am I going wrong?
Best Answer
I don't have a copy of Thomas & Finney to know how it defines limits for functions of two (or more) variables, but some textbooks require that the function be defined at all points in some ball around $(a,b)$, except possibly for $(a,b)$ itself, in order for the limit as $(x,y)\to(a,b)$ to exist. The expression $y\sin(1/x)$ is not defined along the $y$ axis ($x=0$), so in that sense the limit as $(x,y)\to(0,0)$ does not exist. I would suggest looking carefully at your book's definitions.