Question

Find the limit, if it exists.lim _{(x, y)--->(0,0) {x y^{2} cos (y) / {5 x^{2}+y^{4}

Answer 1

To find the limit of the expression (x y^2 cos (y)) / (5 x^2 + y^4) as (x, y) approaches (0, 0), use the Squeeze Theorem and algebraic manipulations to show that the limit is 0.

Step-by-step explanation:

To find the limit of the expression (x y^2 cos (y)) / (5 x^2 + y^4) as (x, y) approaches (0, 0), we can use the Squeeze Theorem and some algebraic manipulations.

1. First, note that the denominator 5x^2 + y^4 is always positive, so we can ignore it when determining the limit.
2. Next, we can rewrite the expression as (y^2 cos(y)) / x. Since x is approaching 0, we can treat it as a constant.
3. Applying the Squeeze Theorem, we know that -1 ≤ cos(y) ≤ 1 for all y. Thus, we have -y^2 ≤ y^2 cos(y) ≤ y^2.
4. Dividing both sides by x, we get -y^2/x ≤ (y^2 cos(y)) / x ≤ y^2/x.
5. As y approaches 0, both -y^2/x and y^2/x go to 0. Therefore, by the Squeeze Theorem, the limit of the expression as (x, y) approaches (0, 0) exists and is 0.