Is my answer to this limit correct?

Question

Define a function $ z(x, y) = \frac{11x^3+5y^3}{x^2+y^2} $. Using the continuous extension, what is the value of the following limit: $$ \lim_{(\Delta x, \Delta y) \rightarrow (0, 0)\\ \text{along the line } x=y \neq 0}{\left(\frac{\Delta z - dz}{\sqrt{(\Delta x)^2+(\Delta y)^2}}\right)} $$ I computed $\Delta z$ and the partial derivatives for $ dz $ and set $x = y$. I got $\Delta z = 8\Delta x$ and $ dz = 8\Delta x$; which gives $ 0 $ as the limit after subtraction. To evaluate $z_{x}$, and $z_{y}$ at the point $ (0, 0) $, I used the following limits: $ \lim_{(x, y) \rightarrow (0, 0)} \left( z_{x}(x, y) \right) $ and $ \lim_{(x, y) \rightarrow (0, 0)} \left( z_{y}(x, y) \right) $ along the line $x=y$. Which give the values $\frac{17}{2}$ and $-\frac{1}{2}$ respectively. If I use the definition of partial derivatives however, I get, 11 and 5 respectively. Along the line $x=y$, the first one gives $dz = 8dx$, and thus the limit is $0$; and the second one gives $dz = 16dx$, and thus the limit is $32$.