Question

Using Squeeze/Sandwich Theorem,

Calculate the limit of : lim [(x,y) -> (0,0)] (3 . x^2 . y / (x^2 + y^2))

Answer 1

Answer: We can use the Squeeze theorem to find the limit.

We know that:

x^2 + y^2 >= 0 for all (x, y)

Also,

|3xy| <= 3|x||y| <= 3x^2 + 3y^2

So,

|3xy/(x^2 + y^2)| <= 3(x^2 + y^2)/(x^2 + y^2) = 3

As we approach (x, y) = (0, 0), x^2 + y^2 becomes arbitrarily close to 0, so the right-hand side of the inequality becomes arbitrarily close to 3, and the left-hand side becomes arbitrarily close to 0.

Therefore,

lim[(x,y)->(0,0)] (3xy/(x^2+y^2)) = 0

Hence, by the sandwich theorem, we can say that the limit of the function is 0.

Explanation: