What value does f (x, y) = (x + 2y) / (x - 2y) approach as (x, y) approaches (0,0) along the x-axis?

A. f (x, y) approaches - 1.

B. f (x, y) has no limit and does not approach infinity or minus infinity as (x, y) approaches (0,0) along the x-axis.

B. f (x, y) has no limit and does not approach infinity or minus infinity as (x, y) approaches (0,0) along the x-axis.

Step-by-step explanation:

Given the function

f (x, y) = (x + 2y) / (x - 2y)

We can apply

y = mx

then

lim (x, y) → (0, 0) f (x, y) = lim (x, mx) → (0, 0) f (x, mx)

⇒ lim (x → 0) (x + 2mx) / (x - 2mx) = lim (x → 0) x (1+2m) / (x * (1-2m)) = (1+2m) / (1-2m)

If

y = x²

then

lim (x, y) → (0, 0) f (x, y) = lim (x → 0) f (x)

⇒ lim (x → 0) (x + 2x²) / (x - 2x²) = (1 + 4 (0)) / (1 - 4 (0)) = 1

If

y = x³

then

lim (x, y) → (0, 0) f (x, y) = lim (x → 0) f (x)

⇒ lim (x → 0) (x + 2x³) / (x - 2x³) = (1 + 6 (0) ²) / (1 - 6 (0) ²) = 1

We applied L'Hopital's rule to solve the limits.

Then, we can say that f (x, y) has no limit since the limits obtained are different.