Question

In Exercises \(57-60,\) find each limit. (a) \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x}\) (b) \(\lim _{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x, y)}{\Delta y}\) \(f(x, y)=\sqrt{y}(y+1)\)

Short answer

(a) 0, (b) Undefined or indeterminate.

Step-by-step solution

Solving Part (a)

Note that \(f(x, y)\) does not depend on \(x\), so \(f(x+\Delta x, y) = f(x, y)\). Therefore, the difference in the numerator of the limit expression is zero, which leads to a limit of zero. So, \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x}=0\).

Solving Part (b) - Step 1: Compute the Difference in the Numerator

The first action is to compute \(f(x, y+\Delta y) - f(x, y)\). Substituting \(y+\Delta y\) into the function yields \(\sqrt{y+\Delta y}((y+\Delta y)+1) = \sqrt{y+\Delta y}(y + \Delta y + 1)\). Therefore, the difference \(f(x, y+\Delta y) - f(x, y) = \sqrt{y+\Delta y}(y + \Delta y + 1) - \sqrt{y}(y+1)\).

Solving Part (b) - Step 2: Compute the Limit

The limit as \( \Delta y\) approaches zero of the difference quotient is computed by substituting \(\Delta y = 0\) into the above expression: \[\lim _{\Delta y \rightarrow 0} \left ( \sqrt{y+\Delta y}(y + \Delta y + 1) - \sqrt{y}(y+1) \right) / \Delta y = \left ( \sqrt{y}(y + 1) - \sqrt{y}(y+1) \right) / 0 = 0 / 0\]. However, applying the limit to a 0/0 form requires L'Hopital's rule or some other method for indeterminate forms, which doesn't apply here. Therefore, the result from part (b) is 'undefined' or 'indeterminate'.