Questions: Find each limit. f(x, y) = sqrt(y)(y+7) (a) lim (Δx -> 0) (f(x+Δx, y) - f(x, y)) / Δx (b) lim (Δy -> 0) (f(x, y+Δy) - f(x, y)) / Δy

Transcript text: Find each limit. \[ f(x, y)=\sqrt{y}(y+7) \] (a) \[ \lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x} \] (b) $\lim _{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x, y)}{\Delta y}$

Solution

Solution Steps

To find the limits in this problem, we need to compute the partial derivatives of the function $ f(x, y) = \sqrt{y}(y+7) $ with respect to $ x $ and $ y $. For part (a), since the function does not explicitly depend on $ x $, the partial derivative with respect to $ x $ will be zero. For part (b), we will compute the partial derivative with respect to $ y $ using the limit definition of a derivative.

Step 1: Function Definition

We start with the function defined as: \[ f(x, y) = \sqrt{y}(y + 7) \]

Step 2: Partial Derivative with Respect to $ x $

To find the limit for part (a): \[ \lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y) - f(x, y)}{\Delta x} \] Since $ f $ does not depend on $ x $, we have: \[ \frac{\partial f}{\partial x} = 0 \]

Step 3: Partial Derivative with Respect to $ y $

For part (b), we compute the limit: \[ \lim_{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y) - f(x, y)}{\Delta y} \] Substituting $ f(y + \Delta y) $: \[ f(y + \Delta y) = \sqrt{y + \Delta y} \left( (y + \Delta y) + 7 \right) \] The partial derivative with respect to $ y $ is given by: \[ \frac{\partial f}{\partial y} = \frac{3y}{2\sqrt{y}} + \frac{7}{2\sqrt{y}} \] This simplifies to: \[ \frac{\partial f}{\partial y} = \frac{3y + 7}{2\sqrt{y}} \]