- #1
Horseboy
- 9
- 0
Homework Statement
Apply the definition of the limit to show that
\begin{align*} f(x,y) = \frac{x^2\,y\,\left( y - 1 \right) ^2 }{x^2 + \left( y-1 \right) ^2 } = 0\end{align*}
I know I'm required to use the epsilon delta method here, no polar stuff either, just straight at it.
Homework Equations
\begin{align*} \sqrt{ \left( x - a \right) ^2 + \left( y - b \right) ^2 } < \delta \implies \left| f(x,y) - L \right| < \epsilon \end{align*}
I can see that a = 0, b = 1 and L = 0.
The Attempt at a Solution
\begin{align*} \sqrt{ \left( x \right) ^2 + \left( y - 1 \right) ^2 } < \delta \implies \left| \frac{x^2y(y-1)^2}{x^2+(y-1)^2} \right| < \epsilon \end{align*}
Which then implies
\begin{align*} x < \delta \end{align*} and \begin{align*} (y-1) < \delta\end{align*}
Now what? Do I sub in delta? Assuming I do, and I guess if (y-1) < delta then y < delta + 1, then I get
\begin{align*} \left| \frac{\delta^3+\delta^2}{2} \right| \end{align*}
But then what? Oh Lordy I'm confused.
Hi all, thanks for checking this thread out! I'm having issues with what to do in general with multivariable limits, but I think I'm getting the hang of it. This is one of the questions I'm having particular trouble with, can anyone offer any advice on what to do from here, or even if I've done it right? Thanks!