- #1

- 38

- 1

**prove that the limit as (x,y) → (0,0) of [(x^2)(siny)^2]/(x^2 + 2y^2) exists**

Here's what I did:

0<√(x^2 + y^2) < δ, |[(x^2)(siny)^2]/(x^2 + 2y^2) - 0| < ε.

x^2 ≤ x^2 + y^2 since y^2 ≥ 0

so it follows that

(x^2)(siny)^2]/(x^2 + 2y^2) ≤ (siny)^2 < ε

siny < y for all y > 0

(siny)^2 < y^2

(siny)^2 < x^2 + y^2 = δ^2

(x^2)(siny)^2]/(x^2 + 2y^2) ≤ (siny)^2 < δ^2 < ε ∴ limit exists at (0,0)

Does this work?