Question

Find the limit, if it exists, or show that the limit does not exist

lim₍ₓ,ᵧ₎→₍₀,₀₎ ​x³+x² y / x²+y²

Answer 1

The limit of the function as (x, y) approaches (0, 0) exists and is equal to 0.

Step-by-step explanation:

To find the limit of the given function, we can approach the point (0, 0) along different paths and see if the function values approach a particular number or if they diverge. Let's consider approaching (0, 0) along the x-axis and y-axis.

When we approach (0, 0) along the x-axis, i.e., setting y = 0, the function becomes lim(x->0) x³ / x² = lim(x->0) x = 0.

When we approach (0, 0) along the y-axis, i.e., setting x = 0, the function becomes lim(y->0) 0 / y² = lim(y->0) 0 = 0.

Since the function values approach the same number (0) along different paths, we can conclude that the limit of the function as (x, y) approaches (0, 0) exists and is equal to 0.