INSTRUCTOR: Show that the limit as x the ordered pair xy approaches the ordered pair 2, 1 of numerator x minus 1 or x minus 2 quantity multiplied by y minus 1 quantity over a denominator of quantity x minus 2 squared plus quantity y minus 1 squared does not exist.

In a question like this, to show that a limit doesn't exist, we have to consider this fact that any path that passes through the point 2, 1 is a path we could actually follow. And for a limit to exist from every direction every path must have the same limit.

So we're going to choose in particular a path of y equals k multiplied by x minus 2 k parentheses x minus 2 close parentheses plus 1. We're choosing that because the point 2,1 is on this path, and it depends on a particular parameter. If the limit depends on the parameter k then the limit doesn't exist. It depends what path you take, then the limit doesn't exist. They all ought to be the same.

So what we are going to do is just note, from this-- this is a very convenient choice that y minus 1 equals k, parentheses x minus 2, close parentheses, which means that y minus 1 squared is equal to k squared, open parenthesis x minus 2, close parentheses squared.

So let's just consider the original function here. So we have x minus 2 in parentheses multiplied by y minus 1, which would be k, parentheses x minus 2, close parentheses. 2 is the numerator divided by x minus 2 in parentheses squared plus, parentheses y minus 1 close parentheses squared, which is k squared parentheses x minus 2 close parentheses squared. Perhaps you can see why we chose this particular parameterized path.

Now, this is going to be equal to, again, that's just replacing y minus 1 y minus 1 squared. Now that is equal to k parentheses x minus 2 close parentheses squared over parentheses 1 plus k squared close parentheses open parenthesis x minus 2 close parentheses squared. Now that reduces to k over 1 plus k squared.

So the limit as xy approaches 2,1 of the original function, numerator parentheses x minus 2 close parentheses open parentheses while minus 1 close parentheses over a denominator of parentheses x minus 2 close parentheses squared plus open parentheses y minus 1 closed prints these squared is equal to k over 1 plus k squared. This limit depends on what k is. It can be very different depending on what you choose for k. So for different paths, we'll have different limits. So the limit does not exist. Limit does not exist.