Question

What value does f(x,y) = x+2y/x-2y approach as (x,y) approaches (0,0) along the x-axis? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. f(x,y) approaches ___(Simplify your answer.)
B. f(x,y) has no limit and does not approach [infinity] or-[infinity] as (x,y) approaches (0,0) along the x-axis.

Answer 1

The limit of the function f(x,y) as (x,y) approaches (0,0) along the x-axis is 1. This is because substituting y = 0 into f(x,y) gives us f(x,0) = x/x, which simplifies to 1 as we approach the point (0,0) along the x-axis.

Step-by-step explanation:

The student is asking about the limit of the function f(x,y) = (x + 2y) / (x - 2y) as the point (x, y) approaches (0, 0) along the x-axis. When approaching (0,0) along the x-axis, y is constantly 0. If we substitute y = 0 into the function, we get f(x, 0) = x / x, which simplifies to 1 for all x not equal to 0. However, we cannot simply put x = 0 into this equation since it would lead to division by zero, which is undefined.

So, as we approach the point (0,0) along the x-axis, where y=0, f(x,y) approaches 1. Therefore, the correct choice for the student's question is:

A. f(x,y) approaches 1 (Simplify your answer.)

Why is it so? As one moves closer to (0,0) along the x-axis, the value of the function becomes increasingly like evaluating f(x,0), which equals 1 for any non-zero x. Hence, the limit of f(x,y) as (x,y) approaches (0,0) along the x-axis is 1.