Show that the limit that follows does not exist by calculating the limit along the given paths. \[ \lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} y}{x^{4}+y^{2}} \] Along Path \#1 $\rightarrow x=0$ : Along Path \#2 $\rightarrow y=0$ : Along Path \#3 $\rightarrow y=x^{2}$ : The limit does not exist because the limits along Path \# and Path \# are not equal.
Solution
Step 1 ：Given the function \(f = \frac{x^{2}y}{x^{4} + y^{2}}\), we are asked to show that the limit as (x, y) approaches (0, 0) does not exist by calculating the limit along the given paths.
Step 2 ：For Path #1, we substitute x=0 into the function and calculate the limit as y approaches 0. This gives us a limit of 0.
Step 3 ：For Path #2, we substitute y=0 into the function and calculate the limit as x approaches 0. This also gives us a limit of 0.
Step 4 ：For Path #3, we substitute y=x^2 into the function and calculate the limit as x approaches 0. This gives us a limit of \(\frac{1}{2}\).
Step 5 ：The limits along Path #1 and Path #2 are both 0, but the limit along Path #3 is \(\frac{1}{2}\). Since the limits along these paths are not equal, the limit of the function as (x, y) approaches (0, 0) does not exist.
Step 6 ：\(\boxed{\text{Final Answer: The limit does not exist because the limits along Path #1 and Path #3 are not equal.}}\)
