Evaluating $\lim_{(x,y) \to (\pi,0)} {\cos(x) + 1 + {y^2/2} \over (x - \pi)^2 + y^2}$ I'm struggling to evaluate this limit problem:
$$\lim\limits_{(x,y) \to (\pi,0)} {\cos x + 1 + {y^2/2} \over (x - \pi)^2 + y^2}$$
I've tried this with paths $x=\pi$, $y=0$, $y=x-\pi$, ... and so far all of them have resulted in $1 \over 2$. However when I try to evaluate this in WolframAlpha, it says that the limit does not exist.
I haven't found any paths that result in a value other than $1/2$. How can I prove that this limit does not exist?

You should try $x' = x - \pi$ and then $$\lim\limits_{(x,y) \to (\pi,0)} {\cos x + 1 + {y^2/2} \over (x - \pi)^2 + y^2} = \lim\limits_{(x',y) \to (0,0)} {1-\cos x' + {y^2/2} \over (x')^2 + y^2}$$
$$=  \lim\limits_{(x,y) \to (0,0)} {1-\cos x \over x^2 + y^2} + \frac{1}{2} {{y^2} \over x^2 + y^2}$$
$$=  \lim\limits_{(x,y) \to (0,0)} { 2\sin^2(x/2) \over x^2 + y^2} + \frac{1}{2} { y^2 \over x^2 + y^2}$$
$$=  \lim\limits_{(x,y) \to (0,0)} \frac{\frac{1}{2}x^2}{x^2+y^2} \left ({\sin^2(x/2)\over (x/2)^2} \right) + {\frac{1}{2}y^2\over x^2 + y^2}$$
$$=  \lim\limits_{(x,y) \to (0,0)} \frac{\frac{1}{2}x^2}{x^2+y^2} \left (1+{\sin^2(x/2)\over (x/2)^2} - 1 \right) + {\frac{1}{2}y^2\over x^2 + y^2}$$
$$=  \lim\limits_{(x,y) \to (0,0)} \frac{1}{2}\frac{x^2 + y^2}{x^2+y^2} + \lim\limits_{(x,y) \to (0,0)} \frac{1}{2}\frac{x^2}{x^2+y^2} \left ( {\sin^2(x/2)\over (x/2)^2} - 1\right )  $$
$$= \frac{1}{2}  $$
