Answer:
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.
