I noticed that this is a big black hole in my understanding of partial derivatives at the point. I don't know how to count it:
$$ f(x,y) = \frac {xy^2}{x^2+y^2} $$
$$ \frac {df}{dx}(0,0)=\lim_{t\to 0} \frac{f(t,0)-f(0,0)}{t} =\lim_{t\to 0} \frac {\frac{t*0}{t^2}}{t} = \frac {0}{0}=??? $$ I don't know how to finish it.
Your solution was correct up to this step: $$ \lim_{t\to 0} \frac {\frac{t*0}{t^2}}{t} = \frac{0}{0}$$ The theorem about limit of quotient being the quotient of limits does not apply when the denominator has limit zero. Therefore, you should continue to transform and simplify the quotient: $$ \lim_{t\to 0} \frac {\frac{t*0}{t^2}}{t} = \lim_{t\to 0} \frac {0/t^2}{t} = \lim_{t\to 0} \frac {0}{t} = \lim_{t\to 0} 0 = 0$$