Hi! I am new and I can't solve this limit:

limit x,y->(0,0) (( x^2 ) * ( y^4)) / (|x^6|+|y^4|)

Can anyone give me a hand?

Try converting to polars:

\(\displaystyle \lim_{(x, y) \to (0,0)}\left(\frac{x^2y^4}{|x^6| + |y^4|}\right) = \lim_{r \to 0}\left[\frac{(r\cos{\theta})^2(r\sin{\theta})^4}{|(r\cos{\theta})^6| + |(r\sin{\theta})^4|}\right]\)

\(\displaystyle = \lim_{r \to 0}\left[\frac{r^6\cos^2{\theta}\sin^4{\theta}}{r^6|\cos^6{\theta}| + r^4|\sin^4{\theta}|}\right]\)

\(\displaystyle = \lim_{r \to 0}\left[\frac{r^2\cos^2{\theta}\sin^4{\theta}}{r^2|\cos^6{\theta}| + |\sin^4{\theta}|}\right]\)

\(\displaystyle = \frac{0^2\cos^2{\theta}\sin^4{\theta}}{0^2|\cos^6{\theta}| + |\sin^4{\theta}|}\)

\(\displaystyle = \frac{0}{|\sin^4{\theta}|}\)

\(\displaystyle = 0\).