So i have the following function, $f:\mathbb{R} ^{2}\rightarrow \mathbb{R}$, and they ask me to analyze the continuity at the point $P = (1,0)$, when f is defined as follows:
$f(x,y) =\begin{cases} \dfrac{\left( x^{2}-2x+1\right) y\cos \left( \dfrac {1} {\left( x-1\right) ^{2}+y^{2}}\right) } {\left( 3\left( x-1\right) ^{2}+y^{2}\right) \left( 2+x^{2}-2x\right) } & \text{ if }(x,y) \neq (1,0)\\ 0 &\text{ if } (x,y) = (1,0)\end{cases}$
So, I know that, in order to be continuous, the limit of $f$ at $(1,0)$ has to be $0$ (the value of the function at the given point). But i don't know how to prove that it's (or it isn't) $0$.
Can you anyone give me a hint of how to face these problem? If you have, any link with similar problems where i can learn the method? Because the type of exercise are very frequent in the exams, but are not developed in the textbook we use (Mardsen and Tromba, Vol. 3).
Thanks a lot for your help!