SUMMARY
The limit of the function f(x,y) = x^2 * sin^2(y) / (x^2 + 2y^2) as (x,y) approaches (0,0) is proven to be 0 using the delta-epsilon method. The key condition to satisfy is |f(x,y) - 0| ≤ ε. It is established that 0 ≤ sin^2(y) ≤ 1, which aids in bounding the function. A suitable function u(δ) must be found such that |f(x,y)| ≤ u(δ) < ε to complete the proof.
PREREQUISITES
- Understanding of delta-epsilon proofs in calculus
- Knowledge of limits in multivariable calculus
- Familiarity with trigonometric functions, specifically sin(y)
- Ability to manipulate inequalities and bounds
NEXT STEPS
- Study the delta-epsilon definition of limits in more detail
- Learn about bounding functions in multivariable calculus
- Explore examples of limits involving trigonometric functions
- Practice solving limit problems using the delta-epsilon method
USEFUL FOR
Students studying calculus, particularly those focusing on multivariable limits, educators teaching limit proofs, and anyone seeking to strengthen their understanding of delta-epsilon arguments.