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) \(\lim _{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x, y)}{\Delta y}\). \(f(x, y)=x^{2}-4 y\)

Short answer

The limit for \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x}\) is \( 2x \), and for \(\lim _{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x, y)}{\Delta y}\), it's \( -4 \).

Step-by-step solution

Replace \( f(x+\Delta x, y) \)

We first replace \( f(x+\Delta x, y) \) in the limit expression. The function \( f(x+\Delta x, y) = (x + \Delta x)^{2}-4y \). Now we get the new expression for limit: \(\lim _{\Delta x \rightarrow 0} \frac{(x + \Delta x)^{2}-4y - f(x, y)}{\Delta x}\)

Simplify (a)

We can replace \( f(x, y) \) with \( x^{2} - 4y \) in the limit we got. After simplifying, we get: \(\lim _{\Delta x \rightarrow 0} \frac{(x+\Delta x)^2 - x^2}{\Delta x}\)

Find Limit (a)

Apply limit properties to the above expression to determine the limit as \( \Delta x \rightarrow 0\). Perform the algebraic operation to get the limit equal to \( 2x \)

Replace \( f(x, y+\Delta y) \)

For part (b), we first replace \( f(x, y+\Delta y) \) in the limit expression. The function \( f(x, y+\Delta y) = x^{2}-4(y+\Delta y) \). We get: \(\lim _{\Delta y \rightarrow 0} \frac{x^{2}-4(y+\Delta y) - f(x, y)}{\Delta y}\)

Simplify (b)

We replace \( f(x, y) \) with \( x^{2} - 4y \) in the limit we got. We get: \(\lim _{\Delta y \rightarrow 0} \frac{-4\Delta y}{\Delta y}\)

Find Limit (b)

Again, apply limit properties to the above expression to determine the limit as \( \Delta y \rightarrow 0\). Perform the algebraic operation to get the limit equal to \( -4 \)