$\lim_{(x,y) \to (0,0)} \frac{{x{y^2}}}{{{x^2} + {y^4}}} = 0$ Please, Anyone could suggest me some way for this?. Thanks.
Prove that $\lim\limits_{(x,y) \to (0,0)} \frac{{x{y^2}}}{{{x^2} + {y^4}}} = 0$
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0See also http://math.stackexchange.com/questions/93886/computing-a-multivariable-limit-lim-x-y-to-0-0-frac2x2yx4-y2 – 2016-10-16
6 Answers
If indeed the limit was zero then every way we approach $(0,0)$ the limit would have to be $0$.
However, if we take the path $x=y^2$ we have:
$\lim_{x\to 0}\frac{x^2}{x^2+x^2}=\frac12\neq 0$
So the limit cannot be zero. Maybe it could be something else, but then it would have to be $\frac12$. Take $y=0$, we have:
$\lim_{x\to 0}\frac0{x^2}=0\neq\frac12$
Therefore the limit does not exist.
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0Great! many thanks. Regards. – 2012-07-23
If $y=x$ the limit is $0$, but if $x=y^2$ the limit is $\frac{1}{2}$, then the limit don't exist.
This is not true. Consider sequence $(x_n,y_n)=(n^{-2},n^{-1})$ then you get $ \lim\limits_{n\to\infty}(x_n,y_n)=0\\ \lim\limits_{n\to\infty}\frac{x_n y_n^2}{x_n^{2}+y_n^{4}}=1 $ If you consider another sequence $(x_n,y_n)=(n^{-1},n^{-1})$ then you get $ \lim\limits_{n\to\infty}(x_n,y_n)=0\\ \lim\limits_{n\to\infty}\frac{x_n y_n^2}{x_n^{2}+y_n^{4}}=+\infty $ So we conclude that limit $ \lim\limits_{(x,y)\to(0,0)}\frac{xy^2}{x^2+y^4} $ even doesn't exist, not to mention it is equal to $0$.
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0Yes, I had not noticed. – 2012-07-23
Let $x=y^2$ therefore we have $\lim\limits_{(x,y)\to(0,0)}\frac{xy^2}{x^2+y^4}=\lim\limits_{y\to0}\frac{y^4}{2y^4}=\frac{1}{2} \ \ \ (1)$ and if we consider $x=y$ then $\lim\limits_{(x,y)\to(0,0)}\frac{xy^2}{x^2+y^4}=0 \ \ \ \ (2)$ so that from (1) and (2) we see that limit don't exits
Let $f(x,y):=\frac{xy^2}{x^2+y^4}$. We have $f(x^2,x)=1/2$ and $f(0,y)=0$, which proves that the limit doesn't exist.
come along any $x=my^2$ then $\displaystyle \lim\limits_{(x,y)\to(0,0)}\frac{xy^2}{x^2+y^4}=\frac{1}{m^2+1}$, so for different $m$ you'll get different limits, so limit does not exists!