Limits of points along X-Axis of the 2-variable function

Question

Let $f:\Re ^2\rightarrow\Re$ be a real-valued function such that f(x,y) is defined as, $$f(x,y)= \begin{cases} xsin(1/y),& \text{if $y\neq 0$} \\[2ex] 0,& \text{if $y=0$} \end{cases}$$ The question is to find $ \lim_{(x,y)\to (a,0)}f(x,y)$,$a\in\Re$.I was able to prove the limit for the point (0,0) is equal to 0.However,for points other than (0,0),the limit does not exist(correct me if I'm wrong) as $sin(1/y)$ oscillates between 1 and -1 as $y\rightarrow 0$ but how do I prove it using $\epsilon$-$\delta$ definition for the limit of a function?Or is there any other way to prove it?

Answer 1

If $f$ had limit in $(a,0)$ it would have limit for every direction you approach $(a, 0)$. Instead, if you go to $(a, 0)$ on the straight line $x=a$ you have $\lim_{y\to 0} f(a, t)=a\lim_{y\to 0} \sin (\frac 1y)$ that doesn't exist.