Question 2

a. Using $\delta$–$\varepsilon$ approach, show that $\displaystyle\lim_{(x,y)\to(0,0)} xy\,\frac{x^2-y^2}{x^2+y^2} = 0$. (2 marks)

b. Check for the continuity of the following functions: (3 marks)

(i) $f(x,y) = \begin{cases} \dfrac{2x^2+y^2}{3+\sin x}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases}$ at $(0,0)$.

(ii) $f(x,y) = \begin{cases} \dfrac{xy}{x^2+2y^2}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases}$ at $(0,0)$.

(iii) $f(x,y) = x^2 + y^2 - 1$, at $(1,1)$.

If continuous, then use $\delta$–$\varepsilon$ approach.