Supremum and Infimum of the multiple variable function

Question

I have a function $$F(t_1,t_2)=\frac{t_1^2a^2+t_2^2 b^2+2t_1 t_2ab}{t_1^2+t_2^2}$$ where $0\leq t_1\leq 1$, $0\leq t_2\leq 1$. I know that there is no limit of the function when $(t_1,t_2)\to (0,0)$. But I want to know, has this function supremum and infimum ?

Answer 1

Hint. Note that for $(t_1,t_2)\not=(0,0)$, $$F(t_1,t_2)=\frac{(t_1a+t_2 b)^2}{t_1^2+t_2^2}.$$ Then by the Cauchy–Schwarz inequality it follows that $$0\leq F(t_1,t_2)\leq \frac{(t_1^2+t_2^2)(a^2+b^2)}{t_1^2+t_2^2}\leq a^2+b^2.$$ Now consider the sequence $(a/n,b/n)\to 0$. What is the limit of $F(a/n,b/n)$?