<s>[INST] Prove or disprove each of the follow function has limits $x \to a$ by the definition $\lim_{(x, y) \to (0,0)}\frac{x^2y}{x^4 + y^2}$ Prove or disprove each of the follow function has limits $x \to a$ by the definition
$$\lim_{(x, y) \to (0,0)}\frac{x^2y}{x^4 + y^2}$$
Let $y = mx^2$
$$\frac{x^2y}{x^4 + y^2}=\frac{x^4m}{x^4(1+m^2)} = \frac{m}{1+m^2}$$
Hence, $$\lim_{(x, y) \to (0,0)}\frac{x^2y}{x^4+y^2} = \frac{m}{1+m^2}$$
Therefore, since it depends on the value of m, the limit does not exist
Correct?
 [/INST] Yes that correct, indeed we can consider for example


*

*for $x=0,\, y\neq 0 \implies \frac{x^2y}{x^4 + y^2}=0$

*for $x=t,\,y=t^2, \,t\to 0 \implies \frac{x^2y}{x^4 + y^2}=\frac{t^4}{t^4 + t^4}\to \frac12$
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