Question

By considering different paths of approach, show that the function below has no limit as (x,y) approaches (0,0).

Answer 1

The function y = 1/x has no limit as (x,y) approaches (0,0) along any path of approach.

Step-by-step explanation:

To show that the function y = 1/x has no limit as (x,y) approaches (0,0), we can consider different paths of approach.

Let's consider approaching (0,0) along the x-axis. As x approaches 0, y = 1/x approaches positive or negative infinity, depending on which side of 0 we approach from. This means that the function does not approach a specific value as (x,y) approaches (0,0) along the x-axis.

Similarly, if we approach (0,0) along the y-axis, as y approaches 0, x = 1/y approaches positive or negative infinity. Therefore, the function does not have a limit as (x,y) approaches (0,0) along the y-axis as well.