Question

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

Answer 1

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.