User:Metajellyfish/Math770/HW7

Compute the iterated limit at $(0,0)$ of the following function. Determine if it has a limit as $(x,y) \to (0,0)$ in $\mathbb{R}^2$, and prove that the limit exists. $$f(x,y) = \frac{x^2 + y^4}{x^2 + 2y^4}$$ We begin by fixing $y$ at $0$. Then: $$\lim_{x \to 0} f(x,0) = \lim_{x \to 0} \frac{x^2}{x^2} = 1$$ Fixing $x = 0$ gives us: $$\lim_{y\to0} f(0,y) = \lim_{x\to 0} \frac{y^4}{2y^4} = \frac{1}{2}$$ Since $\lim_{x\to0} f(x,0) \neq \lim_{x\to0} f(0,y)$, $\lim_{x\to0} f(x,y)$ does not exist.