$$\text { Find the limit } \lim _ { \Delta x \rightarrow 0 } \frac { f ( x + \Delta x , y ) - f ( x , y ) } { \Delta x } \text { for the function } f ( x , y ) = 3 x + 2 x y - 5 y$$
A) $2 x - 5 y$
B) $3 + 2 y$
C) 0
D) $3 + 2 x$
E) $- 5 y$

Question

$$\text { Find the limit } \lim _ { \Delta x \rightarrow 0 } \frac { f ( x + \Delta x , y ) - f ( x , y ) } { \Delta x } \text { for the function } f ( x , y ) = 3 x + 2 x y - 5 y$$ 
A) $2 x - 5 y$
B) $3 + 2 y$
C) 0
D) $3 + 2 x$
E) $- 5 y$

Quizplus · Accepted Answer

To find the partial derivative of $f$ with respect to $x$, we simply treat $y$ as a constant and differentiate with respect to $x$. So,
$$\frac{\partial f}{\partial x} = 3 + 2y$$
Therefore, the limit as $\Delta x 	o 0$ of the expression in the question is just $\frac{\partial f}{\partial x}$ evaluated at $(x,y)$, which is $3 + 2y$. So the answer is $\boxed{	extbf{(B)}}$.

A) 2x−5y2 x - 5 y2x−5y B) 3+2y3 + 2 y3+2y

A) $2 x - 5 y$ B) $3 + 2 y$