SUMMARY
The limit of the expression (cos(xy) - 1) / (x^2 y^2) as (x,y) approaches (0,0) can be solved by multiplying the numerator and denominator by (cos(xy) + 1). This transformation leads to the expression -[sin^2(xy)/(xy)^2] * [1/cos(xy) + 1]. The critical step involves recognizing that the limit of sin(u)/u approaches 1 as u approaches 0, allowing the simplification of sin^2(u)/u^2 to also approach 1, effectively eliminating the (xy)^2 denominator in the limit process.
PREREQUISITES
- Understanding of limits in multivariable calculus
- Familiarity with trigonometric limits, specifically lim(u -> 0) sin(u)/u
- Knowledge of L'Hôpital's Rule for evaluating indeterminate forms
- Basic algebraic manipulation of trigonometric identities
NEXT STEPS
- Study the application of L'Hôpital's Rule in multivariable limits
- Explore the Taylor series expansion for cos(xy) around (0,0)
- Learn about the epsilon-delta definition of limits in calculus
- Investigate other trigonometric limits and their applications in calculus
USEFUL FOR
Students studying calculus, particularly those focusing on multivariable limits, as well as educators seeking to clarify limit evaluation techniques involving trigonometric functions.