Question

By considering different paths of approach, show that the function below has no limit as (x,y)→(0,0). f(x,y)=x^1 /(x^4 +y^2 ) Examine the values of f along curves that end at (0,0). Along which set of curves is f a constant value? A. y=kx ^2 ,x=0 B, y=kx+kx ^2x=0 C. y=kx ^3 ,x=0 D. y=k0,x

=0 If (x,y) approaches (0,0) along the curve when k=1 used in the net of curves found above, what is the limit? (Simplify your answer.) If (x,y) approaches (0,0) along the curve when k=0 used in the set of curves found above, What is the limat? (Simplify your answer) What can you conclude? A. Sinca (has two diferen limits aleng two different paths to (0,0), by the two-path test, fhas no lirnit an (x,y) approaches (0,0) B. Since fhas the same limit along two deferent pathe to (0,0), by the two-path test f has no linsit as (x,y) appreaches (0,0) C. Since f has two different limits along two diferent patus to (0,0). in cannot be determined whether or not f has a Imit as (x,y) approaches (0,0). D. Since f has the same limit along two different paths to (0,0), in cannot be determined whether ar not f has a limit as (x,y) approaches (0,0)

Answer 1

To show that the function f(x,y) = x/(x^4 + y^2) has no limit as (x,y) approaches (0,0), we can examine the values of f along curves that end at (0,0). Along the curve y = kx^3, the function f is a constant value of 0. If (x,y) approaches (0,0) along the curve when k = 1, the limit is 1. If (x,y) approaches (0,0) along the curve when k = 0, the limit is infinity. Therefore, we can conclude that f(x,y) has no limit as (x,y) approaches (0,0) because it has different limits along different paths.

Step-by-step explanation:

In order to show that the function f(x,y) = x/(x^4 + y^2) has no limit as (x,y) approaches (0,0), we need to consider different paths of approach.

Let's examine the values of f along different curves that end at (0,0) to determine if f is a constant value along any set of curves.

A. For the curve y = kx^2, where x is not equal to 0, substituting y into f(x,y), we get f(x,y) = x/(x^4 + k^2x^4) = 1/(x^3(1 + k^2)). Since the value of f depends on x, it is not a constant value along this curve.

B. For the curve y = kx + kx^2, where x is not equal to 0, substituting y into f(x,y), we get f(x,y) = x/(x^4 + (kx + kx^2)^2). Again, since the value of f depends on x, it is not a constant value along this curve.

C. For the curve y = kx^3, where x is equal to 0, substituting y into f(x,y), we get f(x,y) = 0. In this case, f is a constant value along this curve.

Now, if (x,y) approaches (0,0) along the curve when k = 1, substituting this into f(x,y), we get f(x,y) = x/(x^4 + x^2) = 1/(x^2 + 1). As x approaches 0, 1/(x^2 + 1) approaches 1. So, the limit in this case is 1.

If (x,y) approaches (0,0) along the curve when k = 0, substituting this into f(x,y), we get f(x,y) = x/(x^4) = 1/x^3. As x approaches 0, 1/x^3 approaches infinity. So, the limit in this case is infinity.

Based on the above analysis, we can conclude that f(x,y) = x/(x^4 + y^2) has no limit as (x,y) approaches (0,0) since it has different limits depending on the path of approach.