Problem 29 - Calculus Early Transcendentals

Question

Use the Two-Path Test to prove that the following limits do not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{y^{4}-2 x^{2}}{y^{4}+x^{2}}$$

Step-by-Step Solution

Answer

Short answer: Since the limits along the paths y=0 and x=0 are not the same (-2 and 1, respectively), by the Two-Path Test, the limit does not exist when (x, y) approaches (0,0).

Step 1: Approach along the path y=0

Along the path y=0, we substitute y=0 into the function: $$\frac{y^{4}-2x^{2}}{y^{4}+x^{2}} = \frac{(0)^{4}-2x^{2}}{(0)^{4}+x^{2}} = \frac{-2x^{2}}{x^{2}} = -2$$ In this case, the limit as (x, y) approaches (0,0) is -2.

Step 2: Approach along the path x=0

Along the path x=0, we substitute x=0 into the function: $$\frac{y^{4}-2x^{2}}{y^{4}+x^{2}} = \frac{y^{4}-2(0)^{2}}{y^{4}+(0)^{2}} = \frac{y^{4}}{y^{4}} = 1$$ In this case, the limit as (x, y) approaches (0,0) is 1.

Step 3: Compare the limits along different paths

Since the limit is -2 along the path y=0 and 1 along the path x=0, the two limits are not the same. Thus, by the Two-Path Test, the limit does not exist when (x,y) approaches (0,0). Therefore, we conclude that: $$\lim _{(x, y) \rightarrow(0,0)} \frac{y^{4}-2 x^{2}}{y^{4}+x^{2}} \text{ does not exist.}$$